The Ramsey Theory of Henson graphs

TL;DR

This paper proves that for each $k extgreater 3$, the $k$-clique-free Henson graph has finite big Ramsey degrees, extending Ramsey theory to a new class of infinite, ultrahomogeneous graphs using novel coding techniques.

Contribution

The paper introduces a new method of coding Henson graphs into strong coding trees and proves Ramsey theorems for these trees, establishing finite big Ramsey degrees for all $k extgreater 3$.

Findings

Finite big Ramsey degrees are established for all $k extgreater 3$ Henson graphs.

Development of strong coding trees as a new tool for Ramsey theory.

Methodology opens pathways for studying big Ramsey degrees in other ultrahomogeneous structures.

Abstract

Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author's recent result for the triangle-free Henson graph, we prove that for each $k\ge 4$, the $k$-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey's Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker.

Equations

$$C\to (B{)}_j^A$$

$$C\to (B{)}_{j,t(A,\mathcal{K})}^A.$$

$$\forall j\ge 1,\mathcal{S}\to (\mathcal{S}{)}_{j,T(\mathrm{A},\mathcal{S})}^{\mathrm{A}}.$$

$$vEw⟺t(|s|)=1.$$

$$\hat{T}=\{s\in \mathcal{S}:\exists t\in T\exists n\le |t|(s=t\upharpoonright n)\}.$$

$$\text{mbox}{ht}_T(t)=|\{s\in T:s\subset t\}|,$$

$$T(n)=\{t\in T:\text{mbox}{ht}_T(t)=n\}.$$

$$T\upharpoonright l=\{t\in \hat{T}:|t|=l\}.$$

$$s{E}_{\mathcal{G}}t⟺(|s|>|t|\mathrm{and}s(|t|)=1)\mathrm{or}(|t|>|s|\mathrm{and}t(|s|)=1).$$

$$h:\bigcup _{n\in \mathbb{N}}\prod _{i<d}{T}_i(n)\to r$$

$$\bigcup _{n\in \mathbb{N}}\prod _{i<d}{S}_i(n).$$

$$\lambda \to (\kappa {)}_{\sigma }^d$$

$${\beth }_d(\mu {)}^{+}\to ({\mu }^{+}{)}_{\mu }^{d+1}.$$

$$p:d\times {\vec{\delta }}_p\to \bigcup _{i<d}{T}_i\upharpoonright l_p,$$

$${\dot{b}}_{i,\alpha }=\{⟨p(i,\alpha ),p⟩:p\in \mathbb{P},\mathrm{and}(i,\alpha )\in \mathrm{dom}(p)\}.$$

$${\dot{b}}_{\vec{\alpha }}\mathrm{denote}⟨{\dot{b}}_{0,{\alpha }_0},\dots ,{\dot{b}}_{d-1,{\alpha }_{d-1}}⟩,$$

$${\dot{b}}_{\vec{\alpha }}\upharpoonright l\mathrm{denote}\{{\dot{b}}_{i,{\alpha }_i}\upharpoonright l:i<d\}.$$

$$\{\iota (0),\iota (1)\}<\{\iota (2),\iota (3)\}<\cdots <\{\iota (2d-2),\iota (2d-1)\}.$$

$$({\theta }_{\iota (0)},{\theta }_{\iota (2)},\dots ,{\theta }_{\iota (2d-2))}),({\theta }_{\iota (1)},{\theta }_{\iota (3)},\dots ,{\theta }_{\iota (2d-1)}),$$

$$f(\iota ,\vec{\theta })=$$

$$⟨⟨i,j⟩:i<d,j<k_{\vec{\alpha }},{\vec{\delta }}_{\vec{\alpha }}(j)={\alpha }_i⟩,$$

$$⟨⟨j,k⟩:j<k_{\vec{\alpha }},k<k_{\vec{\beta }},{\delta }_{\vec{\alpha }}(j)={\delta }_{\vec{\beta }}(k)⟩⟩.$$

$$p_{\vec{\alpha }}(i,{\alpha }_i)=p_{\vec{\alpha }}(i,{\delta }_{\vec{\alpha }}(j_i))=t_{i,j_i}=t_i^{\ast }.$$

$$p_{\vec{\alpha }}(i,{\delta }_{\vec{\alpha }}(j))=t_{i,j}=p_{\vec{\beta }}(i,{\delta }_{\vec{\beta }}(j)).$$

$$q=\{⟨(i,\delta ),q(i,\delta )⟩:i<d,\delta \in {\vec{\delta }}_q\}.$$

$$T_{\mathrm{G}}=\{t\in \mathcal{S}:\exists n\in N,\exists l\le n(t=c_n\upharpoonright l)\}.$$

$$(\forall j<k-2)t(l_{i_j})=1⟹t(l_n)=0.$$

$$t(l_n)=1⟺n\in F.$$

$$T(m)=\{s\in T:|s|=|d_m^T|\}.$$

$$T=\bigcup _{m\in M}T(m).$$

$$r_m(T)=\bigcup _{j<m}T(j).$$

Full text

We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker.

This research was supported by National Science Foundation Grants DMS-1600781 and DMS-1901753

1. Overview

A central program of the theory of infinite structures is to find which structures have partition properties resembling Ramsey’s Theorem. In this context, one colors the copies of a finite structure $\mathrm{A}$ inside the infinite structure $\mathcal{S}$ into finitely many colors and looks for an infinite substructure $\mathcal{S}^{\prime}$, isomorphic to $\mathcal{S}$, in which the copies of $\mathrm{A}$ have the same color. A wide collection of infinite structures have the Ramsey property for colorings of singletons. However, even the rationals as a linearly ordered structure do not have the Ramsey property for colorings of pairsets, as seen by Sierpiński’s two-coloring of pairs of rationals where each subcopy of the rationals retains both colors on its pairsets. This leads to the following question:

Question 1.1**.**

Given an infinite structure $\mathcal{S}$ and a finite substructure $\mathrm{A}$, is there a positive integer $T(\mathrm{A},\mathcal{S})$ such that for any coloring of all copies of $\mathrm{A}$ in $\mathcal{S}$ into finitely many colors, there is a substructure $\mathcal{S}^{\prime}$ of $\mathcal{S}$, isomorphic to $\mathcal{S}$, in which all copies of $\mathrm{A}$ take no more than $T(\mathrm{A},\mathcal{S})$ colors?

The smallest such number $T(\mathrm{A},\mathcal{S})$, when it exists, is called the big Ramsey degree of $\mathrm{A}$ in $\mathcal{S}$. Research in this area has gained recent momentum, as it was highlighted by Kechris, Pestov, and Todorcevic in [34]. Big Ramsey degrees have implications for topological dynamics, as shown in [34] and further developed in Zucker’s work [57].

In contrast to finite structural Ramsey theory, the development of Ramsey theory for infinite structures has progressed quite slowly. After Sierpiński’s coloring for pairs of rationals, work of Laver and Devlin (see [9]) established the exact big Ramsey degrees for finite sets of rationals by 1979. In the mid 1970’s, Erdős, Hajnal, and Posá began work on the big Ramsey degrees of the Rado graph, establishing an analogue of Sierpiński’s coloring for edges in [22]. Building on work of Pouzet and Sauer in [49], the full Ramsey theory of the Rado graph for colorings of copies of any finite graph was finally established in 2006 in the two papers [52] by Sauer and [37] by Laflamme, Sauer, and Vuksanovic. Around that time, driven by the interest generated by [34], the Ramsey theory of other ultrahomogeneous structures was established in [45] and [36]. A principal component in the work in [9] and [52] is a Ramsey theorem for strong trees due to Milliken [41], while [36] depended on the authors’ development of a colored version of this theorem. The lack of similar means for coding infinite structures and the lack of Ramsey theorems for such coded structures have been the largest obstacles in the further development of this area, especially for ultrahomogeneous structures with forbidden configurations. As stated in Nguyen Van Thé’s habilitation [47], “so far, the lack of tools to represent ultrahomogeneous structures is the major obstacle towards a better understanding of their infinite partition properties.”

In this paper, we prove that for each $k\geq 4$, the $k$-clique-free Henson graph $\mathcal{H}_{k}$ has finite big Ramsey degrees, extending work of the author in [14] for the triangle-free Henson graph. Given $k\geq 3$, the Henson graph $\mathcal{H}_{k}$ is the universal ultrahomogeneous $K_{k}$-free graph; that is, the $k$-clique-free analogue of the Rado graph. The only prior work on the big Ramsey degrees of $\mathcal{H}_{k}$ for $k\geq 4$ was work of El-Zahar and Sauer in [20] for vertex colorings in 1989. In [14], we proved that the triangle-free Henson graph has finite big Ramsey degrees. The work in this paper follows the general outline in [14], but the extension of Ramsey theory to all Henson graphs required expanded ideas, a better understanding of the nature of coding structures with forbidden configurations, and many new lemmas. This article presents a unified framework for the Ramsey theory of Henson graphs. We develop new techniques for coding copies of $\mathcal{H}_{k}$ via strong $\mathcal{H}_{k}$-coding trees and prove Ramsey theorems for these trees, forming a family of Milliken-style theorems. The approach here streamlines the one in [14] for $\mathcal{H}_{3}$ and provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures with and without forbidden configurations.

2. Introduction

The field of Ramsey theory was established by the following celebrated result.

Theorem 2.1** (Infinite Ramsey Theorem, [50]).**

Given positive integers $m$ and $j$, suppose the collection of all $m$-element subsets of $\mathbb{N}$ is colored by $j$ colors. Then there is an infinite set $N$ of natural numbers such that all $m$-element subsets of $N$ have the same color.

From this, Ramsey deduced the following finite version, which also can be proved directly. Throughout the paper, each natural number is identified with the set of its predecessors.

Theorem 2.2** (Finite Ramsey Theorem, [50]).**

Given positive integers $m,n,j$ with $m\leq n$, there is an integer $r>n$ such that for any coloring of the $m$-element subsets of $r$ into $j$ colors, there is a subset $N\subseteq r$ of cardinality $n$ such that all $m$-element subsets of $N$ have the same color.

In both cases, we say that the coloring is monochromatic on $N$. Interestingly, Theorem 2.2 was motivated by Hilbert’s Entscheidungsproblem: to find a decision procedure deciding which formulas in first order logic are valid. Ramsey applied Theorem 2.2 to prove that the validity, or lack of it, for certain types of formulas in first order logic (those with no existential quantifiers) can be ascertained algorithmically. Later, Church and Turing each showed that a general solution to Hilbert’s problem is impossible, so Ramsey’s success for the class of existential formulas is quite remarkable. Ever since the inception of Ramsey theory, its connections with logic have continually spurred progress in both fields. This phenomenon occurs once again in Sections 7 and 8, where methods of logic are used to deduce Ramsey theorems.

Structural Ramsey theory investigates which structures satisfy versions of Ramsey’s Theorem. All structures in this paper are first order, and embeddings are as in model theory. In this setting, one tries to find a substructure isomorphic to some fixed structure on which the coloring is monochromatic. Given structures $\mathrm{A}$ and $\mathrm{B}$, we write $\mathrm{A}\leq\mathrm{B}$ if and only if there is an embedding of $\mathrm{A}$ into $\mathrm{B}$. A substructure $\mathrm{A}^{\prime}$ of $\mathrm{B}$ is called a copy of $\mathrm{A}$ if and only if $\mathrm{A}^{\prime}$ is the image of some embedding of $\mathrm{A}$ into $\mathrm{B}$. The collection of all copies of $\mathrm{A}$ in $\mathrm{B}$ is denoted by ${\mathrm{B}\choose\mathrm{A}}$. Given structures $\mathrm{A},\mathrm{B},\mathrm{C}$ with $\mathrm{A}\leq\mathrm{B}\leq\mathrm{C}$ and an integer $j\geq 1$, we write

[TABLE]

to mean that for each $c:{\mathrm{C}\choose\mathrm{A}}\rightarrow j$, there is a $\mathrm{B}^{\prime}\in{\mathrm{C}\choose\mathrm{B}}$ for which $c$ takes only one color on ${\mathrm{B}^{\prime}\choose\mathrm{A}}$. A class $\mathcal{K}$ of finite structures is said to have the Ramsey property if given $\mathrm{A},\mathrm{B}\in\mathcal{K}$ with $\mathrm{A}\leq\mathrm{B}$, for any integer $j\geq 1$, there is some $\mathrm{C}\in\mathcal{K}$ for which $B\leq C$ and $\mathrm{C}\rightarrow(\mathrm{B})^{\mathrm{A}}_{j}$.

Some classic examples of classes of structures with the Ramsey property include finite Boolean algebras (Graham and Rothschild [29]), finite vector spaces over a finite field (Graham, Leeb, and Rothschild [27] and [28]), finite linearly ordered relational structures (independently, Abramson and Harrington, [1] and Nešetřil and Rödl, [43], [44]), in particular, the class of finite linearly ordered graphs. The papers [43] and [44] further proved that all set-systems of finite linearly ordered relational structures omitting some irreducible substructure have the Ramsey property. This includes the classes of finite linearly ordered graphs omitting $k$-cliques, denoted $\mathcal{G}_{k}^{<}$, for each $k\geq 3$. Fraïssé classes are natural objects for structural Ramsey theory investigations, for as shown by Nešetřil, any class with the Ramsey property must satisfy the amalgamation property. Since Fraïssé theory is not central to the proofs in this article, we refer the interested reader to [24] and Section 2 of the more recent [34] for background; the properties of the specific examples contained in this article will be clear.

Most classes of finite unordered structures do not have the Ramsey property. However, if equipping the class with an additional linear order produces the Ramsey property, then some remnant of it remains in the unordered reduct. This is the idea behind small Ramsey degrees. Following notation in [34], given any Fraïssé class $\mathcal{K}$ of finite structures, for $A\in\mathcal{K}$, $t(A,\mathcal{K})$ denotes the smallest number $t$, if it exists, such that for each $B\in\mathcal{K}$ with $A\leq B$ and for each $j\geq 2$, there is some $C\in\mathcal{K}$ into which $B$ embeds such that for any coloring $c:{C\choose A}\rightarrow j$, there is a $B^{\prime}\in{C\choose B}$ such that the restriction of $c$ to ${B^{\prime}\choose A}$ takes no more than $t$ colors. In the arrow notation, this is written as

[TABLE]

A class $\mathcal{K}$ has finite (small) Ramsey degrees if for each $\mathrm{A}\in\mathcal{K}$ the number $t(\mathrm{A},\mathcal{K})$ exists. The number $t(\mathrm{A},\mathcal{K})$ is called the Ramsey degree of $A$ in $\mathcal{K}$ [23]. Notice that $\mathcal{K}$ has the Ramsey property if and only if $t(A,\mathcal{K})=1$ for each $A\in\mathcal{K}$.

The connection between Fraïssé classes with finite small Ramsey degrees and ordered expansions is made explicit in Section 10 of [34], where it is shown that if an ordered expansion $\mathcal{K}^{<}$ of a Fraïssé class $\mathcal{K}$ has the Ramsey property, then $\mathcal{K}$ has finite small Ramsey degrees. Furthermore, the degree of $\mathrm{A}\in\mathcal{K}$ can be computed from the number of non-isomorphic order expansions it has in $\mathcal{K}^{<}$. Nguyen Van Thé has extended this to the more general notion of pre-compact expansions (see [47]). In particular, the classes of finite (unordered) graphs and finite (unordered) graphs omitting $k$-cliques have finite small Ramsey degrees.

Continuing this expansion of Ramsey theory leads to investigations of which infinite structures have properties similar to Theorem 2.1. Notice that the infinite homogeneous subset $N\subseteq\mathbb{N}$ in Theorem 2.1 is actually isomorphic to $\mathbb{N}$ as a linearly ordered structure. Ramsey theory on infinite structures is concerned with finding substructures isomorphic to the original infinite structure in which a given coloring is as simple as possible. Many infinite structures have been proved to be indivisible: given a coloring of its single-element substructures into finitely many colors, there is an infinite substructure isomorphic to the original structure in which all single-element substructures have the same color. The natural numbers and the rational numbers as linearly ordered structures are indivisible, the proofs being straightforward. Similarly, it is folklore that the Rado graph is indivisible, the proof following naturally from the defining properties of this graph.

In contrast, it took much more effort to prove the indivisibility of the Henson graphs, and this was achieved first for the triangle-free Henson graphs in [35], and for all other Henson graphs in [20]. When one considers colorings of structures of two or more elements, more complexity begins to emerge. Even for the simple structure of the rationals, there is a coloring of pairsets into two colors such that each subset isomorphic to the rationals has pairsets in both colors. This is the infamous example of Sierpiński, and it immediately leads to the notion of big Ramsey degree. We take the definition from [34], slightly changing some notation.

Definition 2.3** ([34]).**

Given an infinite structure $\mathcal{S}$ and a finite substructure $\mathrm{A}\leq\mathcal{S}$, let $T(\mathrm{A},\mathcal{S})$ denote the least integer $T\geq 1$, if it exists, such that given any coloring of ${\mathcal{S}\choose\mathrm{A}}$ into finitely many colors, there is a substructure $\mathcal{S}^{\prime}$ of $\mathcal{S}$, isomorphic to $\mathcal{S}$, such that ${\mathcal{S}^{\prime}\choose\mathrm{A}}$ takes no more than $T$ colors. This may be written succinctly as

[TABLE]

We say that $\mathcal{S}$ has finite big Ramsey degrees if for each finite substructure $\mathrm{A}\leq\mathcal{S}$, there is an integer $T(A,\mathcal{S})\geq 1$ such that (3) holds.

Infinite structures which have been investigated in this light include the rationals ([9]), the Rado graph ([22], [49], [52], [37]), ultrametric spaces ([45]), the rationals with a fixed finite number of equivalence relations, and the tournaments $\bf{S}(2)$ and $\bf{S}(3)$ ([36]), and recently, the triangle-free Henson graph ([14]). These results will be discussed below. See [47] for an overview of results on big Ramsey degrees obtained prior to 2013. Each of these structures is ultrahomogeneous: any isomorphism between two finitely generated substructures can be extended to an automorphism of the infinite structure. Recently, Mašulović has widened the investigation of big Ramsey degrees to universal structures, regardless of ultrahomogeneity, and proved transfer principles in [40] from which big Ramsey degrees for one structure may be transferred to other categorically related structures. More background on the development of Ramsey theory on infinite structures will be given below, but first, we present some recent motivation from topological dynamics for further exploration of big Ramsey degrees.

For the specialist, we briefly remark on connections between topological dynamics and Ramsey theory. These connections have been known for some time (see for instance [26] and [48]) and many of the previously known phenomena were subsumed in the work of Kechris, Pestov, and Todorcevic in [34], where they proved several general correspondences between Ramsey theory and topological dynamics. A Fraïssé  class which has at least one relation which is a linear order is called a Fraïssé order class. One of the main theorems in [34] (Theorem 4.7) shows that the extremely amenable closed subgroups of the infinite symmetric group $S_{\infty}$ are exactly those of the form Aut$(\mathbf{F})$, where $\mathbf{F}$ is the Fraïssé limit (and hence an ultrahomogeneous structure) of some Fraïssé order class satisfying the Ramsey property. Another significant theorem (Theorem 10.8) provides a way to compute the universal minimal flow of topological groups which arise as the automorphism groups of Fraïssé limits of Fraïssé classes with the Ramsey property and the ordering property. That the ordering property can be relaxed to the expansion property was proved by Nguyen Van Thé in [46].

In [34], Kechris, Pestov, and Todorcevic also demonstrated how big Ramsey degrees for Fraïssé  structures $\mathbf{F}$ are related to big oscillation degrees for their automorphism groups, Aut$(\mathbf{F})$. Recently, Zucker proved in [57] that if a Fraïssé structure $\mathbf{F}$ has finite big Ramsey degrees and moreover, $\mathbf{F}$ admits a big Ramsey structure, then any big Ramsey flow of Aut$(\mathbf{F})$ is a universal completion flow, and further, any two universal completion flows are isomorphic.

2.1. A brief history of big Ramsey degrees and main obstacles to its development

In contrast to the robust development for finite structures, results on the Ramsey theory of infinite structures have been meager and the development quite slow. Motivated by Sierpiński’s coloring for pairs of rationals which admits no isomorphic copy in one color, Laver investigated the more general problem of finding whether or not there are bounds for colorings of $m$-sized subsets of rationals, for any positive integer $m$. In the 1970’s, Laver showed that the rationals have finite big Ramsey degrees, finding good upper bounds. Guided by Laver’s results and methods, Devlin found the exact bounds in [9]. Interestingly, these numbers turn out to be coefficients of the Taylor series for the tangent function. Around the same time, Erdős, Hajnal, and Posá initiated investigations of the Rado graph, the universal ultrahomogeneous graph on countably many vertices. In 1975, they proved in [22] that there is a coloring of edges into two colors in which each subcopy of the Rado graph has edges in both colors. That the upper bound for the big Ramsey degree of edges in the Rado graph is exactly two was proved much later (1996) by Pouzet and Sauer in [49]. The full Ramsey theory of the Rado graph was finally established a decade later by Sauer in [52] and by Laflamme, Sauer, and Vuksanovic in [37]. Together, these two papers gave a full description of the big Ramsey degrees of the Rado graph in terms of types of certain trees. A recursive procedure for computing these numbers was given by Larson in [38] soon after. It is notable that while the big Ramsey degrees for the rationals are computed by a closed formula, there is no closed formula producing the big Ramsey degrees for the Rado graph.

The successful completion of the Ramsey theory of the Rado graph so soon after the work of of Kechris, Pestov, and Todorcevic stimulated more interest in Ramsey theory of infinite structures, especially ultrahomogeneous structures, which are obtained as limits of Fraïssé classes. In 2008, Nguyen Van Thé investigated big Ramsey degrees for ultrahomogeneous ultrametric spaces. Given $S$ a set of positive real numbers, $\mathcal{U}_{S}$ denotes the class of all finite ultrametric spaces with strictly positive distances in $S$. Its Fraïssé limit, denoted $\mathbf{Q}_{S}$, is called the Urysohn space associated with $\mathcal{U}_{S}$. In [45], Nguyen Van Thé proved that $\mathbf{Q}_{S}$ has finite big Ramsey degrees whenever $S$ is finite. Moreover, if $S$ is infinite, then any member of $\mathcal{U}_{S}$ of size greater than or equal to $2$ does not have a big Ramsey degree. Soon after this, Laflamme, Nguyen Van Thé, and Sauer proved in [36] that enriched structures of the rationals, and two related directed graphs, have finite big Ramsey degrees. For each $n\geq 1$, $\mathbb{Q}_{n}$ denotes the structure $(\mathbb{Q},Q_{1},\dots,Q_{n},<)$, where $Q_{1},\dots,Q_{n}$ are disjoint dense subsets of $\mathbb{Q}$ whose union is $\mathbb{Q}$. This is the Fraïssé limit of the class $\mathcal{P}_{n}$ of all finite linear orders equipped with an equivalence relation with $n$ many equivalence classes. Laflamme, Nguyen Van Thé, and Sauer proved that each member of $\mathcal{P}_{n}$ has a finite big Ramsey degree in $\mathbb{Q}_{n}$. Further, using the bi-definability between $\mathbb{Q}_{n}$ and the circular directed graphs $\mathbf{S}(n)$, for $n=2,3$, they proved that $\mathbf{S}(2)$ and $\mathbf{S}(3)$ have finite big Ramsey degrees. Central to these results is a colored verision of Milliken’s theorem which they proved in order to deduce the big Ramsey degrees. For more details, we recommend the paper [47] containing a good overview of these results.

A common theme emerges when one looks at the proofs in [9], [52], and [36]. The first two rely in an essential way on Milliken’s Theorem, (see Theorem 3.5 in Section 3). The third proves a new colored version of Milliken’s Theorem and uses it to deduce the results. The results in [45] use Ramsey’s theorem. This would lead one to conclude or at least conjecture that, aside from Ramsey’s Theorem itself, Milliken’s Theorem contains the core combinatorial content of big Ramsey degree results, at least for binary relational structures. The lack of useful representations and lack of Milliken-style theorems for infinite structures in general pose the two main obstacles to broader investigations of big Ramsey degrees. Upon the author’s initial interest in the Ramsey theory of the triangle-free Henson graph, these two challenges were pointed out to the author by Todorcevic in 2012 and by Sauer in 2013; this idea is also expressed in [47], quoted in the Overview.

2.2. Big Ramsey degrees for Henson graphs: Main theorem and prior results

For $k\geq 3$, the Henson graph $\mathcal{H}_{k}$ is the universal ultrahomogeneous $k$-clique free graph. These graphs were first constructed by Henson in 1971 in [32]. It was later noticed that $\mathcal{H}_{k}$ is isomorphic to the Fraïssé limit of the Fraïssé class of finite $k$-clique free graphs, $\mathcal{G}_{k}$. Henson proved in [32] that these graphs are weakly indivisible; given a coloring of the vertices into two colors, either there is a subgraph isomorphic to $\mathcal{H}_{k}$ in which all vertices have the first color, or else every finite $k$-clique free graph has a copy whose vertices all have the second color. However, the indivisibility of $\mathcal{H}_{k}$ took longer to prove. In 1986, Komjáth and Rödl proved in [35] that given a coloring of the vertices of $\mathcal{H}_{3}$ into finitely many colors, there is an induced subgraph isomorphic to $\mathcal{H}_{3}$ in which all vertices have the same color. A few years later, El-Zahar and Sauer proved more generally that $\mathcal{H}_{k}$ is indivisible for each $k\geq 4$ in [20].

Prior to the author’s work in [14], the only further progress on big Ramsey degrees for Henson graphs was for edge relations on the triangle-free Henson graph. In 1998, Sauer proved in [51] that the big Ramsey degree for edges in $\mathcal{H}_{3}$ is two. There, progress stalled for lack of techniques. This intrigued the author for several reasons. Sauer and Todorcevic each stated to the author that the heart of the problem was to find the correct analogue of Milliken’s Theorem applicable to Henson graphs. This would of course help solve the problem of whether Henson graphs have finite big Ramsey degrees, but it would moreover have broader repercussions, as Ramsey theorems for trees are combinatorially strong objects and Milliken’s Theorem has already found numerous applications. The problem was that it was unclear to experts what such an analogue of Milliken’s Theorem should be.

In [14], the author developed an analogue of Milliken’s Theorem applicable to the triangle-free Henson graph, and used it to prove that $\mathcal{H}_{3}$ has finite big Ramsey degrees. In this paper, we provide a unified approach to the Ramsey theory of the $k$-clique-free Henson graphs $\mathcal{H}_{k}$, for each $k\geq 3$. This includes the development of new types of trees which code Henson graphs and new Milliken-style theorems for these classes of trees which are applied to determine upper bounds for the big Ramsey degrees. Our presentation encompasses and streamlines work in [14] for $\mathcal{H}_{3}$. New obstacles arose for $k\geq 4$; these challenges and their solutions are discussed as the sections of the paper are delineated below.

2.3. Outline of paper

In Section 3, we introduce basic definitions and notation, and review strong trees and the Halpern-Läuchli and Milliken Theorems (Theorems 3.8 and 3.5) and their use in obtaining upper bounds for the finite big Ramsey degrees of the Rado graph. In Subsection 3.4, we include a version of Harrington’s forcing proof of the Halpern-Läuchli Theorem. Many new issues arise due to $k$-cliques being forbidden in Henson graphs, but the proof of Theorem 3.8 will at least provide the reader with a toehold into the proof strategies for Theorem 7.2 and Lemma 8.8, which lead to Theorem 8.3, an analogue of Milliken’s Theorem.

The article consists of three main phases. Phase I occurs in Sections 4–6, where we define the tree structures and prove extension lemmas. In Section 4, we introduce the notion of strong $K_{k}$-free trees, analogues of Milliken’s strong trees capable of coding $k$-clique free graphs. These trees contain certain distinguished nodes, called coding nodes, which code the vertices of a given graph. These trees branch maximally, subject to the constraint of the coding nodes not coding any $k$-cliques, and thus are the analogues of strong trees for the $K_{k}$-free setting. Model-theoretically, such trees are simply coding all (incomplete) $1$-types over initial segments of a Henson graph, where the vertices are indexed by the natural numbers. Although it is not possible to fully develop Ramsey theory on strong $K_{k}$-free trees (see Example 4.22), they have the main structural aspects of the trees for which we will prove analogues of Halpern-Läuchli and Milliken Theorems, defined in Section 5. Section 4 is given for the sole purpose of building the reader’s understanding of strong $\mathcal{H}_{k}$-coding trees.

Section 5 presents the definition of strong $\mathcal{H}_{k}$-coding trees as subtrees of a given tree $\mathbb{T}_{k}$ which are strongly isomorphic (Definition 5.10) to $\mathbb{T}_{k}$. The class of these trees is denoted $\mathcal{T}_{k}$, and these trees are best thought of as skew versions of the trees presented in Section 4. Secondarily, an internal description of the trees in $\mathcal{T}_{k}$ is given. This is a new and simpler approach than the one we took in [14] for the triangle-free Henson graph. An important property of strong $\mathcal{H}_{k}$-coding trees is the Witnessing Property (Definition 5.16). This means that certain configurations which can give rise to codings of pre-cliques (Definition 5.7) are witnessed by coding nodes. The effect is a type of book-keeping to guarantee when finite trees can be extended within a given tree $T\in\mathcal{T}_{k}$ to another tree in $\mathcal{T}_{k}$.

Section 6 presents Extension Lemmas, guaranteeing when a given finite tree can be extended to a desired configuration. For $k\geq 4$, some new difficulties arise which did not exist for $k=3$. The lemmas in this section extend work in [14], while addressing new complexities. Further, this section includes some new extension lemmas not in [14]. These have the added benefit of streamlining proofs in Section 7 in which analogues of the Halpern-Läuchli Theorem are proved.

Phase II of the paper takes place in Sections 7 and 8, the goal being to prove the Milliken-style Theorem 8.3 for colorings of certain finite trees, namely those with the Strict Witnessing Property (see Definition 8.1). First, we prove analogues of the Halpern-Läuchli Theorem for strong $\mathcal{H}_{k}$-coding trees in Theorem 7.2. The proof builds on ideas from Harrington’s forcing proof of the Halpern-Läuchli Theorem, but now we must use distinct forcings for two cases: the level sets being colored have either a coding node or else a splitting node. The Extension Lemmas from Section 6 and the Witnessing Property are essential to these proofs. A new ingredient for $k\geq 4$ is that all pre-$a$-cliques for $3\leq a\leq k$ need to be considered and witnessed, not just pre-$k$-cliques. It is important to note that the technique of forcing is used to conduct unbounded searches for finite objects; the proof takes place entirely within the standard axioms of set theory and does not involve passing to a generic model.

In Section 8, we apply induction and fusion lemmas and a third Harrington-style forcing argument to obtain our first Ramsey Theorem for colorings of finite trees.

Theorem 8.3.

Let $k\geq 3$ be given and let $T\in\mathcal{T}_{k}$ be a strong $\mathcal{H}_{k}$-coding tree and let $A$ be a finite subtree of $T$ satisfying the Strict Witnessing Property. Then for any coloring of the copies of $A$ in $T$ into finitely many colors, there is a strong $\mathcal{H}_{k}$-coding tree $S$ contained in $T$ such that all copies of $A$ in $S$ have the same color.

Phase III of the article takes place in Sections 9 and 10. There, we prove a Ramsey theorem for finite antichains of coding nodes (Theorem 9.9), which is then applied to deduce that each Henson graph has finite big Ramsey degrees. To do this, we must first develop a way to transform antichains of coding nodes into finite trees with the Strict Witnessing Property. This is accomplished in Subsections 9.1 and 9.2, where we develop the notions of incremental new pre-cliques and envelopes. Given any finite $K_{k}$-free graph $\mathrm{G}$, there are only finitely many strict similarity types (Definition 9.4) of antichains coding $\mathrm{G}$. Given a coloring $c$ of all copies of $\mathrm{G}$ in $\mathcal{H}_{k}$ into finitely many colors, we transfer the coloring to the envelopes of copies of $\mathrm{G}$ in a given strong coding tree $T$. Then we apply the results in previous sections to obtain a strong $\mathcal{H}_{k}$-coding tree $T^{\prime}\leq T$ in which all envelopes encompassing the same strict similarity type have the same color. Upon thinning to an incremental strong subtree $S\leq T^{\prime}$ while simultaneously choosing a set $W\subseteq T^{\prime}$ of witnessing coding nodes, each finite antichain $X$ of nodes in $S$ is incremental and has an envelope comprised of nodes from $W$ satisfying the Strict Witnessing Property. Applying Theorem 8.3 finitely many times, once for each strict similarity type of envelope, we obtain our second Ramsey theorem for strong $\mathcal{H}_{k}$-coding trees, extending the first one.

Theorem 9.9 (Ramsey Theorem for Strict Similarity Types).

Fix $k\geq 3$. Let $Z$ be a finite antichain of coding nodes in a strong $\mathcal{H}_{k}$-coding tree $T$, and let $h$ be a coloring of all subsets of $T$ which are strictly similar to $Z$ into finitely many colors. Then there is an incremental strong $\mathcal{H}_{k}$-coding tree $S$ contained in $T$ such that all subsets of $S$ strictly similar to $Z$ have the same $h$ color.

Upon taking an antichain of coding nodes $D\subseteq S$ coding $\mathcal{H}_{k}$, the only sets of coding nodes in $D$ coding a given finite $K_{k}$-free graph $\mathrm{G}$ are automatically antichains which are incremental. Applying Theorem 9.9 to the finitely many strict similarity types of antichains coding $\mathrm{G}$, we arrive at the main theorem.

Theorem 10.2.

The universal homogeneous $k$-clique free graph has finite big Ramsey degrees.

For each $\mathrm{G}\in\mathcal{G}_{k}$, the number $T(\mathrm{G},\mathcal{G}_{k})$ is bounded by the number of strict similarity types of antichains of coding nodes coding $\mathrm{G}$. It is presently open to see whether this number is in fact the lower bound. If so, then recent work of Zucker in [57] would provide an interesting connection with topological dynamics, as the colorings obtainable from our structures cohere in the manner necessary to apply Zucker’s work.

*Acknowledgements. *The author would like to thank Andy Zucker for careful reading of a previous version of this paper, pointing out an oversight which led the author to consider singleton pre-cliques when $k\geq 4$, an issue that does not arise in the triangle-free Henson graph. Much gratitude also goes to Dana Bartošová and Jean Larson for listening to early and later stages of these results and for useful feedback; Norbert Sauer for discussing key aspects of the homogeneous triangle-free graph with me during a research visit in Calgary in 2014; Stevo Todorcevic for pointing out to her in 2012 that any proof of finite big Ramsey degrees for $\mathcal{H}_{3}$ would likely involve a new type of Halpern-Läuchli Theorem; and to the organizers and participants of the Ramsey Theory Doccourse at Charles University, Prague, 2016, for their encouragement. The author would like to thank the referee for helpful suggestions which improved this paper. Most of all, the author is grateful for and much indebted to Richard Laver for providing for her in 2011 the main points of Harrington’s forcing proof of the Halpern-Läuchli Theorem, opening the path of applying forcing methods to develop Ramsey theory on infinite structures.

3. Background and the Milliken and Halpern-Läuchli Theorems

This section provides background and sets some notation and terminology for the paper. We review Milliken’s Ramsey Theorem for trees and its application to proving that the Rado graph has finite big Ramsey degrees. Then we discuss why this theorem is not sufficient for handling Henson graphs. In Subsection 3.4, we present the Halpern-Läuchli Theorem, which is a Ramsey theorem on products of trees. This theorem forms the basis for Milliken’s Theorem. We present a version of Harrington’s forcing proof of the Halpern-Läuchli Theorem in order to offer the reader an introduction to the tack we take in later sections toward proving Theorem 8.3.

3.1. Coding graphs via finite binary sequences

The following notation shall be used throughout. The set of all natural numbers $\{0,1,2,\dots\}$ is denoted by $\mathbb{N}$. Each natural number $n$ is equated with the set of all natural numbers strictly less than $n$; thus, $n=\{0,\dots,n-1\}$ and, in particular, [math] is the emptyset. It follows that for $m,n\in\mathbb{N}$, $m<n$ if and only if $m\in n$. For $n\in\mathbb{N}$, we shall let $\mathcal{S}_{n}$ denote the collection of all binary sequences of length $n$. Thus, each $s\in\mathcal{S}_{n}$ is a function from $n$ into $\{0,1\}$, and we often write $s$ as $\langle s(0),\dots,s(n-1)\rangle$. For $i<n$, $s(i)$ denotes the $i$-th value or entry of the sequence $s$. The length of $s$, denoted $|s|$, is the domain of $s$. Note that $\mathcal{S}_{0}$ contains just the empty sequence, denoted $\langle\rangle$.

Notation 3.1**.**

We shall let $\mathcal{S}$ denote $\bigcup_{n\in\mathbb{N}}\mathcal{S}_{n}$, the collection of all binary sequences of finite length.

For nodes $s,t\in\mathcal{S}$, we write $s\subseteq t$ if and only if $|s|\leq|t|$ and for each $i<|s|$, $s(i)=t(i)$. In this case, we say that $s$ is an initial segment of $t$, or that $t$ extends $s$. If $s$ is an initial segment of $t$ and $|s|<|t|$, then we write $s\subset t$ and say that $s$ is a proper initial segment of $t$. For $i\in\mathbb{N}$, we let $s\restriction i$ denote the sequence $s$ restricted to domain $i$. Thus, if $i<|s|$, then $s\restriction i$ is the proper initial segment of $s$ of length $i$, $s\restriction i=\langle s(0),\dots,s(i-1)\rangle$; if $i\geq|s|$, then $s\restriction i$ equals $s$. Let $\mathcal{S}_{\leq n}$ denote $\bigcup_{i\leq n}\mathcal{S}_{i}$, the set of all binary sequences of length at most $n$.

In [22], Erdős, Hajnal and Pósa used the edge relation on a given ordered graph to induce a lexicographic order on the vertices, which they employed to solve problems regarding strong embeddings of graphs. With this lexicographic order, vertices in a given graph can can be represented by finite sequences of [math]’s and $1$’s, a view made explicit in [51] which we review now, using terminology from [52].

Definition 3.2** ([52]).**

Given nodes $s,t\in\mathcal{S}$ with $|s|<|t|$, we call the integer $t(|s|)$ the passing number of $t$ at $s$. Passing numbers represent the edge relation in a graph as follows: Two vertices $v,w$ in a graph can be represented by nodes $s,t\in\mathcal{S}$ with $|s|<|t|$, respectively, if

[TABLE]

Using this correspondence between passing numbers and the edge relation $E$, any graph can be coded by nodes in a binary tree as follows. Let $\mathrm{G}$ be a graph with $N$ vertices, where either $N\in\mathbb{N}$ or $N=\mathbb{N}$, and let $\langle v_{n}:n\in N\rangle$ be any enumeration of the vertices of $\mathrm{G}$. Choose any node $t_{0}\in\mathcal{S}$ to represent the vertex $v_{0}$. For $n>0$, given nodes $t_{0},\dots,t_{n-1}$ in $\mathcal{S}$ coding the vertices $v_{0},\dots,v_{n-1}$, take $t_{n}$ to be any node in $\mathcal{S}$ such that $|t_{n}|>|t_{n-1}|$ and for all $i<n$, $v_{n}$ and $v_{i}$ have an edge between them if and only if $t_{n}(|t_{i}|)=1$. Then the set of nodes $\{t_{n}:n\in N\}$ codes the graph $\mathrm{G}$. For the purposes of developing the Ramsey theory of Henson graphs, we make the convention that the nodes in a tree used to code the vertices in a graph have different lengths. Figure 1.  shows a set of nodes $\{t_{0},t_{1},t_{2},t_{3}\}$ from $\mathcal{S}$ coding the four-cycle $\{v_{0},v_{1},v_{2},v_{3}\}$.

3.2. Trees and related notation

In Ramsey theory on trees, it is standard to use the slightly relaxed definition of tree below. (See for instance Chapter 6 of [54].) Recall that the meet of two nodes $s$ and $t$ in $\mathcal{S}$, denoted $s\wedge t$, is the maximal initial segment of both $s$ and $t$. In particular, if $s\subseteq t$ then $s\wedge t=s$. A set of nodes $A\subseteq\mathcal{S}$ is closed under meets if $s\wedge t$ is in $A$, for every pair $s,t\in A$.

Definition 3.3** (Tree).**

A subset $T\subseteq\mathcal{S}$ is a tree if $T$ is closed under meets and for each pair $s,t\in T$ with $|s|\leq|t|$, $t\restriction|s|$ is also in $T$.

Thus, in our context, a tree $T$ is not necessarily closed under initial segments in $\mathcal{S}$, but rather is closed under initial segments having length in $L(T)=\{|s|:s\in T\}$. Given a tree $T\subseteq\mathcal{S}$, let

[TABLE]

Thus, $\widehat{T}$ is closed under initial segments in the usual sense.

Given a tree $T\subseteq\mathcal{S}$, for each node $t\in T$, define

[TABLE]

where we recall that $s\subset t$ denotes that $s$ is a proper initial segment of $t$. For $n\in\mathbb{N}$, define

[TABLE]

A set $X\subseteq T$ is called a level set if $X\subseteq T(n)$ for some $n\in\mathbb{N}$. Given $l\in\mathbb{N}$, let

[TABLE]

If $l$ is in $L(T)$, then $T\restriction l$ is a level subset of $T$. In this case, there is some $n$ such that $T\restriction l=T(n)$.

3.3. Milliken’s Theorem, its use in proving upper bounds for big Ramsey degrees of the Rado graph, and its insufficiency for Henson graphs

A Ramsey theorem for colorings of finite subtrees of a given tree was proved by Milliken in 1979 (Theorem 3.5 below). This theorem has turned out to be central to proving upper bounds for big Ramsey degrees of several types ultrahomogeneous structures, including the rationals as a linearly ordered structure in [9], the Rado graph and other simple binary relational structures in [52] and [37], and circular directed graphs and rationals with finitely many equivalence relations $\mathbb{Q}_{n}$ in [36]. A good review of these results appears in [47]. Chapter 6 of [54] provides a solid foundation for understanding how Milliken’s theorem is used to deduce upper bounds on big Ramsey degrees for both the rationals and the Rado graph.

Given a tree $T\subseteq\mathcal{S}$, a node $t\in T$ splits in $T$ if and only if there are $u,v\in T$ properly extending $t$ with $u(|t|)=0$ and $v(|t|)=1$.

Definition 3.4** (Strong tree).**

Given a tree $T\subseteq\mathcal{S}$, recall that $L(T)=\{|s|:s\in T\}$, the set of lengths of nodes in $T$. We say that $T$ is a strong tree if and only if for each $t\in T$ with $|t|\neq\sup L(T)$, $t$ splits in $T$. We say that $T$ is a strong tree of height $k$, or simply a $k$-strong tree, if and only if $L(T)$ has exactly $k$ members.

Note that a strong tree $T$ has no maximal nodes if and only if $L(T)$ is infinite. Further, for each $k$-strong subtree of $\mathcal{S}$ is isomorphic as a tree to some binary tree of height $k$, where the isomorphism preserves relative lengths of nodes. In particular, a $1$-strong tree is simply a node in $\mathcal{S}$. See Figure 2. for an example of a $3$-strong tree $T$ with $L(T)=\{0,2,5\}$.

In [41], Milliken proved a Ramsey theorem for finite strong subtrees of finitely branching trees with no maximal nodes. Here, we present a restricted version of that theorem relevant to this paper.

Theorem 3.5** (Milliken, [41]).**

Let $k\geq 1$ be given and let all $k$-strong subtrees of $\mathcal{S}$ be colored by finitely many colors. Then there is an infinite strong subtree $T$ of $\mathcal{S}$ such that all $k$-strong subtrees of $T$ have the same color.

Remark 3.6*.*

A theorem stronger than Theorem 3.5, also due to Milliken in [42], shows that the collection of all infinite strong subtrees of an infinite finitely branching tree forms a topological Ramsey space, meaning that it satisfies an infinite-dimensional Ramsey theorem for Baire sets when equipped with its version of the Ellentuck topology (see [54]). This fact informed some of our intuition when approaching the present work.

Now, we present the basic ideas behind the upper bounds for the big Ramsey degrees of the Rado graph. Upper bounds for the rationals are similar. The work in [36] relied on a stronger variation of Milliken’s theorem proved in that paper; given that theorem, the basic idea behind the upper bounds is similar to what we now present.

The Rado graph, denoted by $\mathcal{R}$, is universal for all countable graphs. However, it is not the only universal countable graph. The following graph $\mathcal{G}$ is also universal. Let $\mathcal{G}$ denote the graph with $\mathcal{S}$ as its countable vertex set and the edge relation $E_{\mathcal{G}}$ defined as follows: For $s,t\in\mathcal{S}$,

[TABLE]

It turns out that this graph $\mathcal{G}$ is also universal, so $\mathcal{G}$ embeds into $\mathcal{R}$ and vice versa.

Suppose that $\mathrm{A}$ is a finite graph and all copies of $\mathrm{A}$ in $\mathcal{R}$ are colored into finitely many colors. Take a copy of $\mathcal{G}$ in $\mathcal{R}$ and restrict our attention to those copies of $\mathrm{A}$ in $\mathcal{G}$. Each copy of $\mathrm{A}$ in $\mathcal{G}$ has vertices which are nodes in $\mathcal{S}$. There are only finitely many strong similarity types of embeddings of $\mathrm{A}$ into $\mathcal{S}$, (Definition 3.1 in [52]), which we now review.

Definition 3.7** (Strong Similarity Map, [52]).**

Let $k\geq 3$ be given and let $S,T\subseteq\mathcal{S}$ be meet-closed subsets. A function $f:S\rightarrow T$ is a strong similarity map from $S$ to $T$ if for all nodes $s,t,u,v\in S$, the following hold:

1. (1)

$f$ is a bijection. 2. (2)

$f$ preserves lexicographic order: $s<_{\mathrm{lex}}t$ if and only if $f(s)<_{\mathrm{lex}}f(t)$. 3. (3)

$f$ preserves meets, and hence splitting nodes: $f(s\wedge t)=f(s)\wedge f(t)$. 4. (4)

$f$ preserves relative lengths: $|s\wedge t|<|u\wedge v|$ if and only if $|f(s)\wedge f(t)|<|f(u)\wedge f(v)|$. 5. (5)

$f$ preserves initial segments: $s\wedge t\subseteq u\wedge v$ if and only if $f(s)\wedge f(t)\subseteq f(u)\wedge f(v)$. 6. (6)

$f$ preserves passing numbers: If $|s|<|t|$, then $f(t)(|f(s)|)=t(|s|)$.

We say that $S$ and $T$ are strongly similar, and write $S\stackrel{{\scriptstyle s}}{{\sim}}T$, exactly when there is a strong similarity map between $S$ and $T$.

The relation $\stackrel{{\scriptstyle s}}{{\sim}}$ is an equivalence relation, and given a fixed finite graph $\mathrm{A}$, there are only finitely many different equivalence classes of strongly similar copies of $\mathrm{A}$ in $\mathcal{G}$. Each equivalence class is called a strong similarity type. Thus, each copy of $\mathrm{A}$ in $\mathcal{G}$ is in exactly one of finitely many strong similarity types.

Fix one strong similarity type for $\mathrm{A}$, call it $\tau$. For each copy $\mathrm{B}$ of $\mathrm{A}$ in $\mathcal{G}$ of type $\tau$, as the vertices of $\mathrm{B}$ are nodes in the tree $\mathcal{S}$, we let $T_{\mathrm{B}}$ denote the tree induced by the meet-closure in $\mathcal{S}$ of the vertices in $\mathrm{B}$; let $k$ be the number of levels of $T_{\mathrm{B}}$. There are finitely many $k$-strong subtrees of $\mathcal{S}$ which contain $T_{\mathrm{B}}$. Moreover, each $k$-strong subtree of $\mathcal{S}$ contains $T_{\mathrm{B}}$ for exactly one $\mathrm{B}$ in $\tau$. (A proof of this fact can be found in Section 6 of [54].) Define a coloring $h$ on the $k$-strong subtrees of $\mathcal{S}$ as follows: Given a $k$-strong subtree $T\subseteq\mathcal{S}$, let $\mathrm{B}$ be the unique copy of $\mathrm{A}$ in $\tau$ such that $T_{\mathrm{B}}$ is contained in $T$. Let $h(T)$ be the color of $\mathrm{B}$. Applying Milliken’s Theorem 3.5, we obtain an infinite strong subtree $S_{\tau}$ of $\mathcal{S}$ with all of its $k$-strong subtrees having the same color. Thus, all copies of $\mathrm{A}$ in $\tau$ with vertices in $S_{\tau}$ have the same color.

Repeating this argument for each strict similarity type, after finitely many applications of Milliken’s Theorem, we obtain an infinite strong subtree $S\subseteq\mathcal{S}$ with the following property: For each strong similarity type $\tau$ for $\mathrm{A}$, all copies of $\mathrm{A}$ of type $\tau$ with vertices in $S$ have the same color. Since $S$ is an infinite strong subtree of $\mathcal{S}$, the subgraph $\mathcal{G}^{\prime}$ of $\mathcal{G}$ coded by the nodes in $S$ is isomorphic to $\mathcal{G}$. Since $\mathcal{R}$ embeds into $\mathcal{G}$, we may take a copy of $\mathcal{R}$ whose nodes come from $S$; call this copy $\mathcal{R}^{\prime}$. Then every copy of $\mathrm{A}$ in $\mathcal{R}^{\prime}$ has the color of its strong similarity type in $S$. The number of strong similarity types for copies of $\mathrm{A}$ is the upper bound for the number of colors of copies of $\mathrm{A}$ in $\mathcal{R}^{\prime}$.

To find the exact big Ramsey degrees, Laflamme, Sauer and Vuksanovic use an additional argument in [37]. We do not reproduce their argument here, as the lower bounds for big Ramsey degrees of the Henson graphs are not the subject of this article. However, we do point out that at the end of the article [52], Sauer takes an antichain $\mathcal{A}$ of nodes in $S$ coding a copy of $\mathcal{R}$ with the further properties: (a) The tree induced by the meet-closure of the nodes in $\mathcal{A}$ has at each level at most one splitting node or one maximal node, but never both. (b) Passing numbers at splitting nodes are always zero, except of course for the right extension of the splitting node itself. These crucial properties (a) and (b) were used to reduce the upper bounds found in [52] to the number of strong similarity types of copies of a finite graph $\mathrm{A}$ occuring in the copy $\mathcal{A}$ of $\mathcal{R}$. This number was then proved to be the exact lower bound for the big Ramsey degrees in [37].

Milliken’s Theorem is not able to handle big Ramsey degrees of Henson graphs for the following reasons: First, there is no natural way to code a Henson graph using all nodes in a strong subtree of $\mathcal{S}$, nor is there a nicely defined graph which is bi-embeddable with a Henson graph. Second, even if there were, there is no way to guarantee that the strong subtree obtained by Milliken’s Theorem would contain a Henson graph. Thus, the need for a new Milliken-style theorem able to handle $k$-clique-free graphs.

We begin with the properties (a) and (b) in mind when we construct trees with special distinguished nodes which code Henson graphs. The reader will notice that our strong $\mathcal{H}_{k}$-coding trees in Section 5.2 have the property that each level of the tree has at most one splitting node or one coding node (Definition 4.1). While our $\mathcal{H}_{k}$-coding trees are certainly not antichains (there are no maximal nodes), they set the stage for taking an antichain of coding nodes which code $\mathcal{H}_{k}$ and have the properties (a) and (b) (Lemma 10.1). This serves to reduce the upper bound on the number of colors. We conjecture that the upper bounds found in this article - the number of strict similarity types of incremental antichains of coding nodes - are in fact the big Ramsey degrees.

3.4. Halpern-Läuchli Theorem and Harrington’s forcing proof

An important Ramsey theorem for trees is Theorem 3.8, due to Halpern and Läuchli. This theorem was found as a key step in the celebrated result of Halpern and Lévy in [31], proving that the Boolean Prime Ideal Theorem (the statement that any filter can be extended to an ultrafilter) is strictly weaker than the Axiom of Choice, assuming the Zermelo-Fraenkel axioms of set theory.

The Halpern-Läuchli Theorem is a Ramsey theorem for colorings of products of level sets of finitely many trees, forming the basis for Milliken’s Theorem 3.5, discussed in the previous subsection. In-depth presentations and proofs of various versions of the Halpern-Läuchli Theorem as well as Milliken’s Theorem can be found in [55], [54], and [17]. The book [55] contains the first published version of a proof due to Harrington using the method of forcing to produce a result inside the standard axioms of set theory, Zermelo-Fraenkel + Axiom of Choice. Harrington’s novel approach is central to the methods we developed in [14] for the triangle-free Henson graph and the more general approach developed in this paper for all Henson graphs. To provide the reader with a warm-up for our proof of Theorem 8.3, we reproduce here a forcing proof from [13]. This proof was outlined for us in 2011 by Richard Laver. It is simpler than the one given in [55] (at the expense of using $\kappa=\beth_{2d-1}(\aleph_{0})^{+}$ instead of the $\kappa=\beth_{d-1}(\aleph_{0})^{+}$ used in [55]) and provides the starting point towards obtaining Theorem 8.3.

The Halpern-Läuchli Theorem holds for finitely many finitely branching trees with no maximal nodes; here, we restrict attention to binary trees since they are sufficient for applications to graphs. The following is the simplest version of the Halpern-Läuchli Theorem for strong trees, which provides the reader with a basic understanding of the starting point for our Ramsey theorems in Sections 7 and 8. Recall that for a tree $T\subseteq\mathcal{S}$, $L(T)$ denotes the set of lengths of nodes in $T$ and $T(n)$ is a level set.

Theorem 3.8** (Halpern-Läuchli, [30]).**

Let $d\geq 1$ be fixed, and for each $i<d$, let $T_{i}$ denote $\mathcal{S}$, the tree of all binary sequences of finite length. Suppose

[TABLE]

is a given coloring, where $r$ is any positive integer. Then there are infinite strong subtrees $S_{i}\subseteq T_{i}$, where $L(S_{i})=L(S_{j})$ for all $i<j<d$, such that $h$ is monochromatic on

[TABLE]

Harrington’s proof uses a cardinal $\kappa$ large enough to satisfy a partition relation guaranteed by the following theorem. Recall that given cardinals $d,\sigma,\kappa,\lambda$, $[\lambda]^{d}$ denotes the collection of all subsets of $\lambda$ of cardinality $d$, and

[TABLE]

means that for each coloring of $[\lambda]^{d}$ into $\sigma$ many colors, there is a subset $X$ of $\lambda$ such that $|X|=\kappa$ and all members of $[X]^{d}$ have the same color. The following is a ZFC result guaranteeing cardinals large enough to have the Ramsey property for colorings into infinitely many colors.

Theorem 3.9** (Erdős-Rado).**

For any non-negative integer $d$ and infinite cardinal $\mu$,

[TABLE]

**Proof of Theorem 3.8. **It is sufficient to consider the case $r=2$. Let $h:\bigcup_{n\in\mathbb{N}}\prod_{i<d}T_{i}(n)\rightarrow 2$ be given. Notice that since each $T_{i}=\mathcal{S}$, it follows that each $T_{i}(n)=T_{i}\restriction n=\mathcal{S}_{n}$. Let $\kappa=\beth_{2d-1}(\aleph_{0})^{+}$. (Recall that $\beth_{1}(\aleph_{0})=2^{\aleph_{0}}$ and in general, $\beth_{n+1}(\aleph_{0})=2^{\beth_{n}(\aleph_{0})}$.) Define $\mathbb{P}$ to be the set of functions $p$ of the form

[TABLE]

where

1. (i)

$\vec{\delta}_{p}$ is a finite subset of $\kappa$ and $l_{p}\in\mathbb{N}$; 2. (ii)

for each $i<d$, $\{p(i,\delta):\delta\in\vec{\delta}_{p}\}\subseteq T_{i}\restriction l_{p}$.

The partial ordering on $\mathbb{P}$ is inclusion: $q\leq p$ if and only if $l_{q}\geq l_{p}$, $\vec{\delta}_{q}\supseteq\vec{\delta}_{p}$, and for each $(i,\delta)\in d\times\vec{\delta}_{p}$, $q(i,\delta)\supseteq p(i,\delta)$.

Forcing with $\mathbb{P}$ adds $\kappa$ branches through the tree $T_{i}$, for each $i<d$. For $\alpha<\kappa$, let $\dot{b}_{i,\alpha}$ denote the $\alpha$-th generic branch through $T_{i}$. Thus,

[TABLE]

Note that for each $p\in\mathbb{P}$ with $(i,\alpha)\in\operatorname{dom}(p)$, $p$ forces that $\dot{b}_{i,\alpha}\restriction l_{p}=p(i,\alpha)$. Let $\dot{\mathcal{U}}$ be a $\mathbb{P}$-name for a non-principal ultrafilter on $\mathbb{N}$. To simplify notation, we write sets $\{\alpha_{i}:i<d\}$ in $[\kappa]^{d}$ as vectors $\vec{\alpha}=\langle\alpha_{0},\dots,\alpha_{d-1}\rangle$ in strictly increasing order. For $\vec{\alpha}=\langle\alpha_{0},\dots,\alpha_{d-1}\rangle\in[\kappa]^{d}$, we let

[TABLE]

and for any $l\in\mathbb{N}$, let

[TABLE]

The goal now is to find disjoint infinite sets $K_{i}\subseteq\kappa$, for $i<d$, and a set of conditions $\{p_{\vec{\alpha}}:\vec{\alpha}\in\prod_{i<d}K_{i}\}$ which are compatible, have the same images in $T$, and such that for some fixed $\varepsilon^{*}$, for each $\vec{\alpha}\in\prod_{i<d}K_{i}$, $p_{\vec{\alpha}}$ forces $h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon^{*}$ for $\dot{\mathcal{U}}$-many $l$. Moreover, we will find nodes $t^{*}_{i}$, $i\leq d$, such that for each $\vec{\alpha}\in\prod_{i<d}K_{i}$, $p_{\vec{\alpha}}(i,\alpha_{i})=t^{*}_{i}$. These will serve as the basis for the process of building the strong subtrees $S_{i}\subseteq T_{i}$ on which $h$ is monochromatic.

For each $\vec{\alpha}\in[\kappa]^{d}$, choose a condition $p_{\vec{\alpha}}\in\mathbb{P}$ such that

1. (1)

$\vec{\alpha}\subseteq\vec{\delta}_{p_{\vec{\alpha}}}$; 2. (2)

$p_{\vec{\alpha}}\Vdash$ “There is an $\varepsilon\in 2$ such that $h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon$ for $\dot{\mathcal{U}}$ many $l$”; 3. (3)

$p_{\vec{\alpha}}$ decides a value for $\varepsilon$, label it $\varepsilon_{\vec{\alpha}}$; and 4. (4)

$h(\{p_{\vec{\alpha}}(i,\alpha_{i}):i<d\})=\varepsilon_{\vec{\alpha}}$.

Such conditions $p_{\vec{\alpha}}$ may be obtained as follows. Given $\vec{\alpha}\in[\kappa]^{d}$, take $p_{\vec{\alpha}}^{1}$ to be any condition such that $\vec{\alpha}\subseteq\vec{\delta}_{p_{\vec{\alpha}}^{1}}$. Since $\mathbb{P}$ forces $\dot{\mathcal{U}}$ to be an ultrafilter on $\mathbb{N}$, there is a condition $p_{\vec{\alpha}}^{2}\leq p_{\vec{\alpha}}^{1}$ such that $p_{\vec{\alpha}}^{2}$ forces that $h(\dot{b}_{\vec{\alpha}}\restriction l)$ is the same color for $\dot{\mathcal{U}}$ many $l$. Furthermore, there must be a stronger condition deciding which of the colors $h(\dot{b}_{\vec{\alpha}}\restriction l)$ takes on $\dot{\mathcal{U}}$ many levels $l$. Let $p_{\vec{\alpha}}^{3}\leq p_{\vec{\alpha}}^{2}$ be a condition which decides this color, and let $\varepsilon_{\vec{\alpha}}$ denote that color. Finally, since $p_{\vec{\alpha}}^{3}$ forces that for $\dot{\mathcal{U}}$ many $l$ the color $h(\dot{b}_{\vec{\alpha}}\restriction l)$ will equal $\varepsilon_{\vec{\alpha}}$, there is some $p_{\vec{\alpha}}^{4}\leq p_{\vec{\alpha}}^{3}$ which decides some level $l$ so that $h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon_{\vec{\alpha}}$. If $l_{p_{\vec{\alpha}}^{4}}<l$, let $p_{\vec{\alpha}}$ be any member of $\mathbb{P}$ such that $p_{\vec{\alpha}}\leq p_{\vec{\alpha}}^{4}$ and $l_{p_{\vec{\alpha}}}=l$. If $l_{p_{\vec{\alpha}}^{4}}\geq l$, let $p_{\vec{\alpha}}=\{((i,\delta),p_{\vec{\alpha}}^{4}(i,\delta)\restriction l):(i,\delta)\in d\times\vec{\delta}_{p_{\vec{\alpha}}^{4}}\}$, the truncation of $p_{\vec{\alpha}}^{4}$ to images that have length $l$. Then $p_{\vec{\alpha}}$ forces that $\dot{b}_{\vec{\alpha}}\restriction l=\{p_{\vec{\alpha}}(i,\alpha_{i}):i<d\}$, and hence $p_{\vec{\alpha}}$ forces that $h(\{p_{\vec{\alpha}}(i,\alpha_{i}):i<d\})=\varepsilon_{\vec{\alpha}}$.

Recall that we chose $\kappa$ large enough so that $\kappa\rightarrow(\aleph_{1})^{2d}_{\aleph_{0}}$ holds. Now we prepare for an application of the Erdős-Rado Theorem. Given two sets of ordinals $J,K$ we shall write $J<K$ if and only if every member of $J$ is less than every member of $K$. Let $D_{e}=\{0,2,\dots,2d-2\}$ and $D_{o}=\{1,3,\dots,2d-1\}$, the sets of even and odd integers less than $2d$, respectively. Let $\mathcal{I}$ denote the collection of all functions $\iota:2d\rightarrow 2d$ such that

[TABLE]

Thus, each $\iota$ codes two strictly increasing sequences $\iota\restriction D_{e}$ and $\iota\restriction D_{o}$, each of length $d$. For $\vec{\theta}\in[\kappa]^{2d}$, $\iota(\vec{\theta}\,)$ determines the pair of sequences of ordinals

[TABLE]

both of which are members of $[\kappa]^{d}$. Denote these as $\iota_{e}(\vec{\theta}\,)$ and $\iota_{o}(\vec{\theta}\,)$, respectively. To ease notation, let $\vec{\delta}_{\vec{\alpha}}$ denote $\vec{\delta}_{p_{\vec{\alpha}}}$, $k_{\vec{\alpha}}$ denote $|\vec{\delta}_{\vec{\alpha}}|$, and let $l_{\vec{\alpha}}$ denote $l_{p_{\vec{\alpha}}}$. Let $\langle\delta_{\vec{\alpha}}(j):j<k_{\vec{\alpha}}\rangle$ denote the enumeration of $\vec{\delta}_{\vec{\alpha}}$ in increasing order.

Define a coloring $f$ on $[\kappa]^{2d}$ into countably many colors as follows: Given $\vec{\theta}\in[\kappa]^{2d}$ and $\iota\in\mathcal{I}$, to reduce the number of subscripts, letting $\vec{\alpha}$ denote $\iota_{e}(\vec{\theta}\,)$ and $\vec{\beta}$ denote $\iota_{o}(\vec{\theta}\,)$, define

[TABLE]

Let $f(\vec{\theta}\,)$ be the sequence $\langle f(\iota,\vec{\theta}\,):\iota\in\mathcal{I}\rangle$, where $\mathcal{I}$ is given some fixed ordering. Since the range of $f$ is countable, applying the Erdős-Rado Theorem 3.9, there is a subset $K\subseteq\kappa$ of cardinality $\aleph_{1}$ which is homogeneous for $f$. Take $K^{\prime}\subseteq K$ such that between each two members of $K^{\prime}$ there is a member of $K$. Take subsets $K_{i}\subseteq K^{\prime}$ such that $K_{0}<\dots<K_{d-1}$ and each $|K_{i}|=\aleph_{0}$.

Lemma 3.10**.**

There are $\varepsilon^{*}\in 2$, $k^{*}\in\mathbb{N}$, and $\langle t_{i,j}:j<k^{*}\rangle$, $i<d$, such that $\varepsilon_{\vec{\alpha}}=\varepsilon^{*}$, $k_{\vec{\alpha}}=k^{*}$, and $\langle p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j)):j<k_{\vec{\alpha}}\rangle=\langle t_{i,j}:j<k^{*}\rangle$, for each $i<d$, for all $\vec{\alpha}\in\prod_{i<d}K_{i}$.

Proof.

Let $\iota$ be the member in $\mathcal{I}$ which is the identity function on $2d$. For any pair $\vec{\alpha},\vec{\beta}\in\prod_{i<d}K_{i}$, there are $\vec{\theta},\vec{\theta}^{\prime}\in[K]^{2d}$ such that $\vec{\alpha}=\iota_{e}(\vec{\theta}\,)$ and $\vec{\beta}=\iota_{e}(\vec{\theta}^{\prime}\,)$. Since $f(\iota,\vec{\theta}\,)=f(\iota,\vec{\theta}^{\prime}\,)$, it follows that $\varepsilon_{\vec{\alpha}}=\varepsilon_{\vec{\beta}}$, $k_{\vec{\alpha}}=k_{\vec{\beta}}$, and $\langle\langle p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j)):j<k_{\vec{\alpha}}\rangle:i<d\rangle=\langle\langle p_{\vec{\beta}}(i,\delta_{\vec{\beta}}(j)):j<k_{\vec{\beta}}\rangle:i<d\rangle$. ∎

Let $l^{*}$ denote the length of the nodes $t_{i,j}$.

Lemma 3.11**.**

Given any $\vec{\alpha},\vec{\beta}\in\prod_{i<d}K_{i}$, if $j,j^{\prime}<k^{*}$ and $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(j^{\prime})$, then $j=j^{\prime}$.

Proof.

Let $\vec{\alpha},\vec{\beta}$ be members of $\prod_{i<d}K_{i}$ and suppose that $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(j^{\prime})$ for some $j,j^{\prime}<k^{*}$. For each $i<d$, let $\rho_{i}$ be the relation from among $\{<,=,>\}$ such that $\alpha_{i}\,\rho_{i}\,\beta_{i}$. Let $\iota$ be a member of $\mathcal{I}$ such that for each $\vec{\zeta}\in[K]^{2d}$ and each $i<d$, $\zeta_{\iota(2i)}\ \rho_{i}\ \zeta_{\iota(2i+1)}$. Take $\vec{\theta}\in[K^{\prime}]^{2d}$ satisfying $\iota_{e}(\vec{\theta})=\vec{\alpha}$ and $\iota_{o}(\vec{\theta})=\vec{\beta}$. Since between any two members of $K^{\prime}$ there is a member of $K$, there is a $\vec{\gamma}\in[K]^{d}$ such that for each $i<d$, $\alpha_{i}\,\rho_{i}\,\gamma_{i}$ and $\gamma_{i}\,\rho_{i}\,\beta_{i}$. Given that $\alpha_{i}\,\rho_{i}\,\gamma_{i}$ and $\gamma_{i}\,\rho_{i}\,\beta_{i}$ for each $i<d$, there are $\vec{\mu},\vec{\nu}\in[K]^{2d}$ such that $\iota_{e}(\vec{\mu})=\vec{\alpha}$, $\iota_{o}(\vec{\mu})=\vec{\gamma}$, $\iota_{e}(\vec{\nu})=\vec{\gamma}$, and $\iota_{o}(\vec{\nu})=\vec{\beta}$. Since $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(j^{\prime})$, the pair $\langle j,j^{\prime}\rangle$ is in the last sequence in $f(\iota,\vec{\theta})$. Since $f(\iota,\vec{\mu})=f(\iota,\vec{\nu})=f(\iota,\vec{\theta})$, also $\langle j,j^{\prime}\rangle$ is in the last sequence in $f(\iota,\vec{\mu})$ and $f(\iota,\vec{\nu})$. It follows that $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\gamma}}(j^{\prime})$ and $\delta_{\vec{\gamma}}(j)=\delta_{\vec{\beta}}(j^{\prime})$. Hence, $\delta_{\vec{\gamma}}(j)=\delta_{\vec{\gamma}}(j^{\prime})$, and therefore $j$ must equal $j^{\prime}$. ∎

For any $\vec{\alpha}\in\prod_{i<d}K_{i}$ and any $\iota\in\mathcal{I}$, there is a $\vec{\theta}\in[K]^{2d}$ such that $\vec{\alpha}=\iota_{o}(\vec{\theta})$. By homogeneity of $f$ and by the first sequence in the second line of equation (19), there is a strictly increasing sequence $\langle j_{i}:i<d\rangle$ of members of $k^{*}$ such that for each $\vec{\alpha}\in\prod_{i<d}K_{i}$, $\delta_{\vec{\alpha}}(j_{i})=\alpha_{i}$. For each $i<d$, let $t^{*}_{i}$ denote $t_{i,j_{i}}$. Then for each $i<d$ and each $\vec{\alpha}\in\prod_{i<d}K_{i}$,

[TABLE]

Lemma 3.12**.**

The set of conditions $\{p_{\vec{\alpha}}:\vec{\alpha}\in\prod_{i<d}K_{i}\}$ is compatible.

Proof.

Suppose toward a contradiction that there are $\vec{\alpha},\vec{\beta}\in\prod_{i<d}K_{i}$ such that $p_{\vec{\alpha}}$ and $p_{\vec{\beta}}$ are incompatible. By Lemma 3.10, for each $i<d$ and $j<k^{*}$,

[TABLE]

Thus, the only way $p_{\vec{\alpha}}$ and $p_{\vec{\beta}}$ can be incompatible is if there are $i<d$ and $j,j^{\prime}<k^{*}$ such that $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(j^{\prime})$ but $p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j))\neq p_{\vec{\beta}}(i,\delta_{\vec{\beta}}(j^{\prime}))$. Since $p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j))=t_{i,j}$ and $p_{\vec{\beta}}(i,\delta_{\vec{\beta}}(j^{\prime}))=t_{i,j^{\prime}}$, this would imply $j\neq j^{\prime}$. But by Lemma 3.11, $j\neq j^{\prime}$ implies that $\delta_{\vec{\alpha}}(j)\neq\delta_{\vec{\beta}}(j^{\prime})$, a contradiction. Therefore, $p_{\vec{\alpha}}$ and $p_{\vec{\beta}}$ must be compatible. ∎

We now construct the strong subtrees $S_{i}\subseteq T_{i}$, for each $i<d$, by induction on the number of levels in the trees. For each $i<d$, let $S_{i}(0)=\{t_{i}^{*}\}$ and let $l_{0}=|t_{i}^{*}|$, the length of the $t_{i}^{*}$, which is well defined since all nodes in the range of any $p\in\mathbb{P}$ have the same length.

Assume now that $n\geq 1$, there are lengths $l_{0}<\dots<l_{n-1}$, and we have constructed finite strong subtrees $\bigcup_{m<n}S_{i}(m)$ of $T_{i}$, $i<d$, such that for each $m<n$, $h$ takes color $\varepsilon^{*}$ on each member of $\prod_{i<d}S_{i}(m)$.

For each $i<d$, let $X_{i}$ denote the set of immediate extensions in $T_{i}$ of the nodes in $S_{i}(n-1)$. For each $i<d$, let $J_{i}$ be a subset of $K_{i}$ with the same size as $X_{i}$. For each $i<d$, label the nodes in $X_{i}$ as $\{q(i,\delta):\delta\in J_{i}\}$. Let $\vec{J}$ denote $\prod_{i<d}J_{i}$. Notice that for each $\vec{\alpha}\in\vec{J}$ and $i<d$, $q(i,\alpha_{i})\supseteq t^{*}_{i}=p_{\vec{\alpha}}(i,\alpha_{i})$.

We now construct a condition $q\in\mathbb{P}$ such that for each $\vec{\alpha}\in\vec{J}$, $q\leq p_{\vec{\alpha}}$. Let $\vec{\delta}_{q}=\bigcup\{\vec{\delta}_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\}$. For each pair $(i,\gamma)$ with $i<d$ and $\gamma\in\vec{\delta}_{q}\setminus J_{i}$, there is at least one $\vec{\alpha}\in\vec{J}$ and some $j^{\prime}<k^{*}$ such that $\delta_{\vec{\alpha}}(j^{\prime})=\gamma$. For any other $\vec{\beta}\in\vec{J}$ for which $\gamma\in\vec{\delta}_{\vec{\beta}}$, since the set $\{p_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\}$ is pairwise compatible by Lemma 3.12, it follows that $p_{\vec{\beta}}(i,\gamma)$ must equal $p_{\vec{\alpha}}(i,\gamma)$, which is exactly $t^{*}_{i,j^{\prime}}$. Let $q(i,\gamma)$ be the leftmost extension of $t_{i,j^{\prime}}^{*}$ in $T$. Thus, $q(i,\gamma)$ is defined for each pair $(i,\gamma)\in d\times\vec{\delta}_{q}$. Define

[TABLE]

Lemma 3.13**.**

For each $\vec{\alpha}\in\vec{J}$, $q\leq p_{\vec{\alpha}}$.

Proof.

Given $\vec{\alpha}\in\vec{J}$, by our construction for each pair $(i,\gamma)\in d\times\vec{\delta}_{\vec{\alpha}}$, we have $q(i,\gamma)\supseteq p_{\vec{\alpha}}(i,\gamma)$. ∎

To construct the $n$-th level of the strong trees $S_{i}$, take an $r\leq q$ in $\mathbb{P}$ which decides some $l_{n}\geq l_{q}$ for which $h(\dot{b}_{\vec{\alpha}}\restriction l_{n})=\varepsilon^{*}$, for all $\vec{\alpha}\in\vec{J}$. By extending or truncating $r$, we may assume without loss of generality that $l_{n}$ is equal to the length of the nodes in the image of $r$. Notice that since $r$ forces $\dot{b}_{\vec{\alpha}}\restriction l_{n}=\{r(i,\alpha_{i}):i<d\}$ for each $\vec{\alpha}\in\vec{J}$, and since the coloring $h$ is defined in the ground model, it is simply true in the ground model that $h(\{r(i,\alpha_{i}):i<d\})=\varepsilon^{*}$ for each $\vec{\alpha}\in\vec{J}$. For each $i<d$ and $\alpha_{i}\in J_{i}$, extend the nodes in $X_{i}$ to length $l_{n}$ by extending $q(i,\delta)$ to $r(i,\delta)$. Thus, for each $i<d$, we define $S_{i}(n)=\{r(i,\delta):\delta\in J_{i}\}$. It follows that $h$ takes value $\varepsilon^{*}$ on each member of $\prod_{i<d}S_{i}(n)$.

For each $i<d$, let $S_{i}=\bigcup_{n\in\mathbb{N}}S_{i}(n)$. Then each $S_{i}$ is a strong subtree of $T_{i}$ with the same set of lengths $L(S_{i})=\{l_{n}:n\in\mathbb{N}\}$, and $h$ takes value $\varepsilon^{*}$ on $\bigcup_{n\in\mathbb{N}}\prod_{i<d}S_{i}(n)$. $\square$

Remark 3.14*.*

This theorem of Halpern and Läuchli was applied by Laver in [39] to prove that given $k\geq 2$ and given any coloring of the product of $k$ many copies of the rationals $\mathbb{Q}^{k}$ into finitely many colors, there are subsets $X_{i}$ of the rationals which again are dense linear orders without endpoints such that $X_{0}\times\dots\times X_{k-1}$ has at most $k!$ colors. Laver further proved that $k!$ is the lower bound. Thus, the big Ramsey degree for the simplest object ($k$-length sequences) in the Fraïssé class of products of $k$-many finite linear orders has been found.

Shelah extended the arguments above, applying forcing methods to prove consistent versions of the Halpern-Läuchli Theorem at a measurable cardinal in [53]. Modifications were used to prove big Ramsey degrees for the $\kappa$-rationals and $\kappa$-Rado graph by Džamonja, Larson, and Mitchell in [18] and [19]. Further work on the Halpern-Läuchli Theorem at uncountable cardinals has been continued in [15], [16] and by Zhang who proved the analogue of Laver’s result [39] for measurable cardinals in [56].

4. Trees coding Henson graphs

This section introduces a unified approach for coding the Henson graphs via trees with special distinguished nodes. We call these trees strong $K_{k}$-free trees (Definition 4.16), since they branch as fully as possible (like strong trees) subject to never coding $k$-cliques. The constructions extend and streamline the construction of strong triangle-free trees in [14]. The distinguished nodes will code the vertices of a Henson graph. The nodes in a given level of a strong $K_{k}$-free tree will code all possible types over the finite graph coded by the lower levels of the tree. Example 3.18 of [14] showed that there is a bad coloring for strong $K_{3}$-free trees which thwarts their development of Ramsey theory. This will be overcome by using skewed versions of strong $K_{k}$-free trees, called strong $\mathcal{H}_{k}$-coding trees, developed in Section 5. The work in the current section provides the reader with the essential structure of and intuition behind strong coding trees utilized in the remainder of the paper.

4.1. Henson’s Criterion

Recall that $K_{k}$ denotes a $k$-clique, a complete graph on $k$ vertices. In [32], for each $k\geq 3$, Henson constructed an ultrahomogeneous $K_{k}$-free graph which is universal for all $K_{k}$-free graphs on countably many vertices. We denote these graphs by $\mathcal{H}_{k}$. It was later seen that $\mathcal{H}_{k}$ is isomorphic to the Fraïssé limit of the Fraïssé class of finite $K_{k}$-free graphs. Given a graph $\mathrm{H}$ and a subset $V_{0}$ of the vertices of $\mathrm{H}$, let $\mathrm{H}|V_{0}$ denote the induced subgraph of $\mathrm{H}$ on the vertices in $V_{0}$. In [32], Henson proved that a countable graph $\mathrm{H}$ is ultrahomogeneous and universal for countable $K_{k}$-free graphs if and only if $\mathrm{H}$ satisfies the following property ($A_{k}$).

1. ($A_{k}$)

2. (i)

$\mathrm{H}$ does not admit any $k$-cliques. 2. (ii)

If $V_{0},V_{1}$ are disjoint finite sets of vertices of $\mathrm{H}$, and $\mathrm{H}|V_{0}$ has no copy of $K_{k-1}$, then there is another vertex which is connected in $\mathrm{H}$ to every member of $V_{0}$ and to no member of $V_{1}$.

The following equivalent modification will be useful for our constructions.

1. $(A_{k})^{\prime}$

2. (i)

$\mathrm{H}$ does not admit any $k$-cliques. 2. (ii)

Let $\langle v_{n}:n\in\mathbb{N}\rangle$ enumerate the vertices of $\mathrm{H}$, and let $\langle F_{i}:i\in\mathbb{N}\rangle$ be any enumeration of the finite subsets of $\mathbb{N}$ such that for each $i\in\mathbb{N}$, $\max(F_{i})<i$, and each finite set appears infinitely many times in the enumeration. Then there is a strictly increasing sequence $\langle n_{i}:i\in\mathbb{N}\rangle$ such that for each $i\in\mathbb{N}$, if $\mathrm{H}|\{v_{m}:m\in F_{i}\}$ has no copy of $K_{k-1}$, then for all $m<i$, $v_{n_{i}}\,E\,v_{m}\longleftrightarrow m\in F_{i}$.

It is routine to check that any countably infinite graph $\mathrm{H}$ is ultrahomogeneous and universal for countable $K_{k}$-free graphs if and only if $(A_{k})^{\prime}$ holds.

4.2. Trees with coding nodes and strong $K_{k}$-free trees

As seen for the case of triangle-free graphs in [14], enriching trees with a collection of distinguished nodes allows for coding graphs with forbidden configurations into trees which have properties similar to strong trees. Recall that $\mathcal{S}$ denotes the set of all finite sequences of [math]’s and $1$’s.

Definition 4.1** ([14]).**

A tree with coding nodes is a structure $(T,N;\subseteq,<,c^{T})$ in the language $\mathcal{L}=\{\subseteq,<,c\}$, where $\subseteq$ and $<$ are binary relation symbols and $c$ is a unary function symbol, satisfying the following: $T$ is a subset of $\mathcal{S}$ satisfying that $(T,\subseteq)$ is a tree (recall Definition 3.3), either $N\in\mathbb{N}$ or $N=\mathbb{N}$, $<$ is the usual linear order on $N$, and $c^{T}:N\rightarrow T$ is an injective function such that whenever $m<n$ in $N$, then $|c^{T}(m)|<|c^{T}(n)|$.

Notation 4.2**.**

The $n$-th coding node in $T$, $c^{T}(n)$, will usually be denoted as $c^{T}_{n}$. The length $|c^{T}_{n}|$ of the $n$-th coding node in $T$ shall be denoted by $l^{T}_{n}$. Whenever no ambiguity arises, we shall drop the superscript $T$.

Throughout this paper, we use $N$ to denote either a member of $\mathbb{N}=\{0,1,2,\dots\}$, or $\mathbb{N}$ itself. We shall treat the natural numbers as von Neumann ordinals. Thus, for $N\in\mathbb{N}$, $N$ denotes the set of natural numbers less than $N$; that is, $N=\{0,\dots,N-1\}$. Hence, in either case that $N\in\mathbb{N}$ or $N=\mathbb{N}$, it makes sense to write $n\in N$.

Definition 4.3** ([14]).**

A graph $\mathrm{G}$ with vertices enumerated as $\langle v_{n}:n\in N\rangle$ is represented by a tree $T$ with coding nodes $\langle c_{n}:n\in N\rangle$ if and only if for each pair $i<n$ in $N$, $v_{n}\,\mathrm{E}\,v_{i}\Longleftrightarrow c_{n}(l_{i})=1$. We will often simply say that $T$ codes $\mathrm{G}$.

For each copy of $\mathcal{H}_{k}$ with vertices indexed by $\mathbb{N}$, there is a tree with coding nodes representing the graph. In fact, this is true more generally for any graph, finite or infinite.

Definition 4.4** (The tree $T_{\mathrm{G}}$ coding $\mathrm{G}$).**

Let $\mathrm{G}$ be any graph with vertices ordered as $\{v_{i}:i\in N\}$. Define $T_{\mathrm{G}}$ as follows: Let $c_{0}=\langle\rangle$, the empty sequence. For $n\geq 1$ with $n\in N$, given coding nodes $\{c_{i}:i<n\}$ coding $\mathrm{G}|\{v_{i}:i<n\}$, with each $|c_{i}|=i$, define $c_{n}$ to be the node in $\mathcal{S}$ of length $n$ such that for each $i<n$, $c_{n}(i)=1\ \longleftrightarrow\ v_{n}\,E\,v_{i}$. Let

[TABLE]

Observation 4.5**.**

Let $\mathrm{G}$ and $T_{\mathrm{G}}$ be as in Definition 4.4. Notice that for each $n\geq 1$ with $n\in N$, each node in $T_{\mathrm{G}}\restriction n$ (the nodes in $T_{\mathrm{G}}$ of length $n$) represents a model-theoretic (incomplete) $1$-type over the graph $\mathrm{G}|\{v_{i}:i<n\}$. Moreover, each such $1$-type is represented by a unique node in $T_{\mathrm{G}}\restriction n$. In particular, if $\mathrm{G}$ is a Rado graph or a Henson graph, then $T_{\mathrm{G}}$ has no maximal nodes and the coding nodes in $T_{\mathrm{G}}$ are dense.

Our goal is to develop a means for working with subtrees of trees like $T_{\mathcal{H}_{k}}$, where $\mathcal{H}_{k}$ is a $k$-clique-free Henson graph, for which we can prove Ramsey theorems like the Halpern-Läuchli Theorem 3.8 and the Milliken Theorem 3.5. There are several reasons why the most naïve approach does not work; these will be pointed out as they arise. In this and the next two sections, we develop tools for recognizing which trees coding $\mathcal{H}_{k}$ and which of their subtrees are able to carry a robust Ramsey theory. These can be interpreted model-theoretically in terms of types over finite subgraphs, but the language of trees will be simpler and easier to visualize.

In Definition 4.4, we showed how to make a tree with coding nodes coding a particular copy of $\mathcal{H}_{k}$; this is a “top-down” approach. To develop Ramsey theory for colorings of finite trees, we will need to consider all subtrees of a given tree $T$ coding $\mathcal{H}_{k}$ which are “similar” enough to $T$ to make a Ramsey theorem possible. In order to prove the Ramsey theorems, we will further need criteria for how and when we can extend a finite subtree $A$ of a given tree $S$, which is a subtree of some $T$, where $T$ codes a copy of $\mathcal{H}_{k}$, to a subtree of $S$ coding another copy of $\mathcal{H}_{k}$. This will provide a “bottom-up” approach for constructing trees coding $\mathcal{H}_{k}$. The potential obstacles are cliques coded by coding nodes in $T$, but which are not coded by coding nodes in $S$. To begin, we observe exactly how cliques are coded.

Observation 4.6**.**

For $a\geq 2$, given an index set $I$ of size $a$, a collection of coding nodes $\{c_{i}:i\in I\}$ codes an $a$-clique if and only if for each pair $i<j$ in $I$, $c_{j}(l_{i})=1$.

Definition 4.7** ($K_{k}$-Free Criterion).**

Let $T\subseteq\mathcal{S}$ be a tree with coding nodes $\langle c_{n}:n<N\rangle$. We say that $T$ satisfies the $K_{k}$-Free Criterion if the following holds: For each $n\geq k-2$, for all increasing sequences $i_{0}<i_{1}<\dots<i_{k-2}=n$ such that $\{c_{i_{j}}:j<k-1\}$ codes a $(k-1)$-clique, for each $t\in T$ such that $|t|>l_{n}$,

[TABLE]

In words, a tree $T$ with coding nodes $\langle c_{n}:n\in N\rangle$ satisfies the $K_{k}$-Free Criterion if for each $n\in N$, whenever a node $t$ in $T$ has the same length as the coding node $c_{n}$, the following holds: If $t$ and $c_{n}$ both code edges with some collection of $k-2$ many coding nodes in $T$ which themselves code a $(k-2)$-clique, then $t$ does not split in $T$; its only allowable extension in $\widehat{T}$ is $t^{\frown}0$.

The next lemma characterizes tree representations of $K_{k}$-free graphs. We say that the coding nodes in $T$ are dense in $T$, if for each $t\in T$, there is some coding node $c_{n}\in T$ such that $t\subseteq c_{n}$. Note that a finite tree $T$ in which the coding nodes are dense will necessarily have coding nodes (of differing lengths) as its maximal nodes.

Lemma 4.8**.**

Let $T\subseteq\mathcal{S}$ be a tree with coding nodes $\langle c_{n}:n\in N\rangle$ coding a countable graph $\mathrm{G}$ with vertices $\langle v_{n}:n\in N\rangle$. Assume that the coding nodes in $T$ are dense in $T$. Then $\mathrm{G}$ is a $K_{k}$-free graph if and only if the tree $T$ satisfies the $K_{k}$-Free Criterion.

Proof.

If $T$ does not satisfy the $K_{k}$-Free Criterion, then there are $i_{0}<\dots<i_{k-2}$ in $N$ and $t\in T$ with $|t|>l_{i_{k-2}}$ such that $\{c_{i_{j}}:j<k-1\}$ codes a $(k-1)$-clique and $t(l_{i_{j}})=1$ for all $j<k-1$. Since the coding nodes are dense in $T$, there is an $n>i_{k-2}$ such that $c_{n}\supseteq t$. Then $\{c_{i_{j}}:j<k-1\}\cup\{c_{n}\}$ codes a $k$-clique. On the other hand, if $\mathrm{G}$ contains a $k$-clique, then there are $i_{0}<\dots<i_{k-1}$ such that the coding nodes $\{c_{i_{j}}:j<k\}$ in $T$ code a $k$-clique, and these coding nodes witness the failure of the $K_{k}$-Free Criterion in $T$. ∎

Definition 4.9** ($K_{k}$-Free Branching Criterion).**

A tree $T$ with coding nodes $\langle c_{n}:n\in N\rangle$ satisfies the $K_{k}$-Free Branching Criterion ($k$-FBC) if $T$ is maximally branching, subject to satisfying the $K_{k}$-Free Criterion.

Thus, $T$ satisfies the $K_{k}$-Free Branching Criterion if and only if $T$ satisfies the $K_{k}$-Free Criterion, and given any $n\in N$ and non-maximal node $t\in T$ of length $l_{n}$, (a) there is a node $t_{0}\in T$ such that $t_{0}\supseteq t^{\frown}0$, and (b) there is a $t_{1}\in T$ such that $t_{1}\supseteq t^{\frown}1$ if and only if for all sequences $i_{0}<\dots<i_{k-2}=n$ such that $\{c_{i_{j}}:j<k-1\}$ codes a copy of $K_{k-1}$, $t(l_{i_{j}})=0$ for at least one $j<k-1$.

As we move toward defining strong $K_{k}$-free trees in Definition 4.16, we recall that the modified Henson criterion $(A_{k})^{\prime}$ is satisfied by an infinite $K_{k}$-free graph if and only if it is ultrahomogeneous and universal for all countable $K_{k}$-free graphs. The following reformulation translates $(A_{k})^{\prime}$ in terms of trees with coding nodes. We say that a tree $T\subseteq\mathcal{S}$ with coding nodes $\langle c_{n}:n\in\mathbb{N}\rangle$ satisfies property $(A_{k})^{\tt tree}$ if the following hold:

1. $(A_{k})^{\tt{tree}}$

2. (i)

$T$ satisfies the $K_{k}$-Free Criterion. 2. (ii)

Let $\langle F_{n}:n\in\mathbb{N}\rangle$ be any enumeration of finite subsets of $\mathbb{N}$ such that for each $n\in\mathbb{N}$, $\max(F_{n})<n-1$, and each finite subset of $\mathbb{N}$ appears as $F_{n}$ for infinitely many indices $n$. Given $n\in\mathbb{N}$, if for each subset $J\subseteq F_{k}$ of size $k-1$, $\{c_{j}:j\in J\}$ does not code a $(k-1)$-clique, then there is some $m\geq n$ such that for all $j<i$, $c_{m}(l_{j})=1$ if and only if $j\in F_{n}$.

Observation 4.10**.**

If $T$ satisfies $(A_{k})^{\tt tree}$, then the coding nodes in $T$ code $\mathcal{H}_{k}$.

To see this, suppose that $T$ satisfies $(A_{k})^{\tt tree}$, and let $\mathcal{H}$ be the graph with vertices $\langle v_{n}:n\in\mathbb{N}\rangle$ where for $m<n$, $v_{n}\ E\ v_{m}$ if and only if $c_{n}(l_{m})=1$. Then $\mathcal{H}$ satisfies Henson’s property $(A_{k})$, and hence is ultrahomogeneous and universal for countable $k$-clique-free graphs.

Observation 4.11**.**

Let $\mathcal{H}$ be a copy of $\mathcal{H}_{k}$ with vertices ordered as $\langle v_{n}:n\in\mathbb{N}\rangle$. Then $T_{\mathcal{H}}$ (recall Definition 4.4) satisfies the $K_{k}$-Free Branching Criterion.

The next lemma shows that any finite tree with coding nodes satisfying the $k$-FBC, where all maximal nodes have height $l_{N-1}$, has the property that every $1$-type over the graph represented by $\{c_{i}:i<N-1\}$ is represented by a maximal node in the tree. This is the vital step toward proving Theorem 4.14: Any tree satisfying the $k$-FBC with no maximal nodes and with coding nodes forming a dense subset codes the $k$-clique-free Henson graph.

Lemma 4.12**.**

Let $T$ be a finite tree with coding nodes $\langle c_{n}:n\in N\rangle$, where $N\geq 1$, satisfying the $K_{k}$-Free Branching Criterion with all maximal nodes of length $l_{N-1}$. Given any $F\subseteq N-1$ for which the set $\{c_{n}:n\in F\}$ codes no $(k-1)$-cliques, there is a maximal node $t\in T$ such that for all $n\in N-1$,

[TABLE]

Proof.

The proof is by induction on $N\geq 1$ over all such trees with $N$ coding nodes. For $N=1$, $N-1=0$, so the lemma trivially holds.

Now suppose $N\geq 2$ and suppose the lemma holds for all trees with less than $N$ coding nodes. Let $T$ be a tree with coding nodes $\langle c_{n}:n\in N\rangle$ satisfying the $k$-FBC. Let $F$ be a subset of $N-1$ such that $\{c_{n}:n\in F\}$ codes no $(k-1)$-cliques. By the induction hypothesis, $\{t\in T:|t|\leq l_{N-2}\}$ satisfies the lemma. So there is a node $t$ in $T$ of length $l_{N-2}$ such that for all $n\in N-2$, $t(l_{n})=1$ if and only if $n\in F\setminus\{N-2\}$. If $N-2\not\in F$, then the maximal node in $T$ extending $t^{\frown}0$ satisfies the lemma; this node is guaranteed to exist by the $k$-FBC.

Now suppose $N-2\in F$. We claim that there is a maximal node $t^{\prime}$ in $T$ which extends $t^{\frown}1$. If not, then $t^{\frown}1$ is not in $\widehat{T}$. By the $k$-FBC, this implies that there is some sequence $i_{0}<\dots<i_{k-2}=N-2$ such that $\{c_{i_{j}}:j<k-1\}$ codes a $(k-1)$-clique and $t(l_{i_{j}})=1$ for each $j<k-2$. Since for all $i<N-2$, $t(l_{i})=1$ if and only if $i\in F\setminus\{N-2\}$, it follows that $\{i_{j}:j<k-2\}\subseteq F$. But then $F\supseteq\{i_{j}:j<k-1\}$, which contradicts that $F$ codes no $(k-1)$-cliques. Therefore, $t^{\frown}1$ is in $\widehat{T}$. Taking $t^{\prime}$ to be maximal in $T$ and extending $t^{\frown}1$ satisfies the lemma. ∎

Remark 4.13*.*

Lemma 4.12 says the following: Suppose $T$ is a tree with coding nodes $\langle c_{n}:n\in N\rangle$ satisfying the $K_{k}$-FBC, $m+1\in N$, and $\mathrm{G}$ is the graph represented by $\langle c_{n}:n\in m\rangle$. Let $G^{\prime}$ be any graph on $m+1$ vertices such that $G^{\prime}\restriction m=G$. Then there is a node $t\in T\restriction l_{m}$ such that letting $c^{\prime}_{m}=t$, the graph represented by $\{c_{n}:n\in m\}\cup\{c^{\prime}_{m}\}$ is isomorphic to $\mathrm{G}^{\prime}$.

Theorem 4.14**.**

Let $T$ be a tree with infinitely many coding nodes satisfying the $K_{k}$-Free Branching Criterion. If $T$ has no maximal nodes and the coding nodes are dense in $T$, then $T$ satisfies $(A_{k})^{\tt tree}$, and hence codes $\mathcal{H}_{k}$.

Proof.

Since $T$ satisfies the $k$-FBC, it automatically satisfies (i) of $(A_{k})^{\tt tree}$. Let $\langle F_{n}:n\in\mathbb{N}\rangle$ be an enumeration of finite subsets of $\mathbb{N}$ where each set is repeated infinitely many times, and each $\max(F_{n})<n-1$. For $n=0$, $F_{n}$ is the emptyset, so every coding node in $T$ fulfills (ii) of $(A_{k})^{\tt tree}$. Let $n\geq 1$ be given and suppose that for each subset $J\subseteq F_{n}$ of size $k-1$, $\{c_{j}:j\in J\}$ does not code a $(k-1)$-clique. By Lemma 4.12, there is some node $t\in T$ of length $l_{n-1}$ such that for all $i<n-1$, $t(l_{i})=1$ if and only if $i\in F_{n}$. Since the coding nodes are dense in $T$, there is some $m\geq n$ such that $c_{m}$ extends $t$. This coding node $c_{m}$ fulfills (ii) of $(A_{k})^{\tt tree}$. ∎

At this point, we have developed enough ideas and terminology to define strong $K_{k}$-free trees. These will be special types of trees coding copies of $\mathcal{H}_{k}$ with additional properties which set the stage for their skew versions in Section 5 on which we will be able to develop a viable Ramsey theory. We shall use ghost coding nodes for the first $k-3$ levels. Coding nodes will start at length $k-2$, and all coding nodes of length at least $k-2$ will end in a sequence of $(k-2)$ many $1$’s. The effect is that coding nodes will only be extendible by [math]; coding nodes will never split. This will serve to reduce the upper bound on the big Ramsey degrees for $\mathcal{H}_{k}$.

Recall that $\mathcal{S}_{\leq n}$ is the set of all sequences of [math]’s and $1$’s of length $\leq n$.

Notation 4.15**.**

Throughout this paper, we use the notation $0^{(i)}$ and $1^{(i)}$ to denote sequences of length $i$ where all entries are [math], or all entries are $1$, respectively.

Definition 4.16** (Strong $K_{k}$-Free Tree).**

A strong $K_{k}$-free tree is a tree with coding nodes, $(T,\mathbb{N};\subseteq,<,c)$, satisfying the following:

1. (1)

$T$ has no maximal nodes, the coding nodes are dense in $T$, and no coding node splits in $T$. 2. (2)

The first $k-2$ levels of $T$ are exactly $\mathcal{S}_{\leq k-2}$, and the least coding node $c_{0}$ is exactly $1^{(k-2)}$. 3. (3)

For each $n\in\mathbb{N}$, the $n$-th coding node $c_{n}$ has length $n+k-2$, and has as final segment a sequence of $k-2$ many $1$’s. 4. (4)

$T$ satisfies the $K_{k}$-free Branching Criterion.

Moreover, $T$ has ghost coding nodes $c_{-k+2},\dots,c_{-1}$ defined by $c_{n}=1^{(k+n-2)}$ for $n\in[-k+2,-1]$, where $1^{(0)}$ denotes the empty sequence. A finite strong $K_{k}$-free tree is the restriction of a strong $K_{k}$-free tree to some finite level.

By Theorem 4.14, each strong $K_{k}$-free tree codes $\mathcal{H}_{k}$.

Remark 4.17*.*

Items (1) and (4) in Definition 4.16 ensure that the tree represents a $K_{k}$-free Henson graph. Items (2) and (3) serve to reduce our bounds on the big Ramsey degrees by ensuring that coding nodes never split. For any node $t$ in $T$ with all entries being [math], the subtree $S$ of all nodes in $T$ extending $t$ codes a copy of $\mathcal{H}_{k}$, by Theorem 4.14. Moreover, the structure of the first $k-2$ levels of such a subtree $S$ are tree isomorphic to $\mathcal{S}_{\leq k-2}$. Thus, it makes sense to require (2) in Definition 4.16. The ghost coding nodes provide the structure which subtrees coding $\mathcal{H}_{k}$ in the same order as $T$ automatically inherit. This will enable us to build the collection of all subtrees of a given tree $T$ which are isomorphic to $T$ in a strong way, to be made precise in the next section.

We now present a method for constructing strong $K_{k}$-free trees. For $k=3$, this construction method simplifies the construction of a strong triangle-free tree coding $\mathcal{H}_{3}$ in Theorem 3.14 of [14] and accomplishes the same goals. The aim of Example 4.18 is to build the reader’s understanding of the principal structural properties of the trees on which we will develop Ramsey theory, before defining their skew versions in the next section.

Example 4.18** (Construction Method for a Strong $K_{k}$-Free Tree, $\mathbb{S}_{k}$).**

Recall that by Theorem 4.14, each strong $K_{k}$-free tree codes $\mathcal{H}_{k}$. Let $\langle u_{i}:i\in\mathbb{N}\rangle$ be any enumeration of $\mathcal{S}$ such that $|u_{i}|\leq|u_{j}|$ whenever $i<j$. Notice that in particular, $|u_{i}|\leq i$. We will build a strong $K_{k}$-free tree $\mathbb{S}_{k}\subseteq\mathcal{S}$ with the $n$-th coding node $c_{n}$ of length $l_{n}=n+k-2$ and satisfying the following additional conditions for $n\geq k-2$:

1. (i)

If $n\equiv 0\ (\mathrm{mod\ }k-1)$, $i=n/(k-1)$, and $u_{i}$ is in $\mathbb{S}_{k}\cap\mathcal{S}_{\leq i+1}$, then $c_{n}\supseteq u_{i}$. 2. (ii)

Otherwise, $c_{n}={0^{(n)}}^{\frown}1^{(k-2)}$.

The first $k-2$ levels of $\mathbb{S}_{k}$ are exactly $\mathcal{S}_{\leq k-2}$. The ghost coding nodes are defined as in Definition 4.16, with $c_{-k+2}$ being the empty sequence and $c_{-1}$ being $1^{(k-3)}$. The shortest coding node is $c_{0}=1^{(k-2)}$. Notice that since $u_{0}=\langle\rangle$, $c_{0}$ extends $u_{0}$.

$\mathbb{S}_{k}$ will have nodes of every finite length, so $\mathbb{S}_{k}(n)=\mathbb{S}_{k}\restriction n$ for each $n\in\mathbb{N}$. We shall let $r_{n}(\mathbb{S}_{k})$ denote $\bigcup_{m<n}\mathbb{S}_{k}(m)$; this notation comes from topological Ramsey space theory in [54] and will be used again in the next section. Then $r_{k-1}(\mathbb{S}_{k})=\mathcal{S}_{\leq k-2}$. Extend the nodes of $\mathbb{S}_{k}(k-2)$ according to the $k$-FBC with respect to $\{c_{-k+2},\dots,c_{0}\}$ to form the next level $\mathbb{S}_{k}(k-1)$. Let $c_{1}=\langle 0\rangle^{\frown}1^{(k-2)}$. This node is in $\mathbb{S}_{k}(k-1)$, since it codes no $k$-cliques with $\{c_{-k+2},\dots,c_{0}\}$. Extend the nodes of $\mathbb{S}_{k}(k-1)$ according to the $k$-FBC with respect to $\{c_{-k+2},\dots,c_{1}\}$ to form the next level $\mathbb{S}_{k}(k)$. So far, (1)–(4) of a Strong $K_{k}$-Free Tree and (i) and (ii) above are satisfied.

Given $n\geq k-1$, suppose we have defined $r_{n+2}(\mathbb{S}_{k})$ and $\{c_{-k+2},\dots,c_{n-k+2}\}$ so that (1)–(4) and (i) and (ii) hold so far. If $n\not\equiv 0\ (\mathrm{mod\ }k-1)$, or $n\equiv 0\ (\mathrm{mod\ }k-1)$ but $u_{i}\not\in r_{i+1}(\mathbb{S}_{k})$, where $i=n/(k-1)$, then define $c_{n}={0^{(n)}}^{\frown}1^{(k-2)}$. This node is in $\mathbb{S}_{k}(n+1)$ by the $k$-FBC, since the only nodes $c_{n}$ codes edges with are exactly the coding nodes $c_{n-k+2},\dots,c_{n-1}$.

Now suppose that $n\equiv 0\ (\mathrm{mod\ }k-1)$ and $u_{i}$ is in $r_{i+1}(\mathbb{S}_{k})$, where $i=n/(k-1)$. Let $q$ denote $n-|u_{i}|$ and define $v={u_{i}}^{\frown}{0^{(q)}}^{\frown}1^{(k-2)}$. We claim that $v$ is in $\mathbb{S}_{k}(n+1)$. Let $-k+2\leq m_{0}<\dots<m_{k-2}\leq n-1$ and suppose that $\{c_{m_{j}}:j\in k-1\}$ is a set of coding nodes in $r_{n}(\mathbb{S}_{k})$ coding a $(k-1)$-clique. If $m_{k-2}<|u_{i}|$, then $v(m_{j})=0$ for at least one $j\in k-1$, since $u_{i}$ is in $r_{|u_{i}|+1}(\mathbb{S}_{k})$ which satisfies the $k$-FBC. Let $w={u_{i}}^{\frown}{0^{(q)}}$. If for some $j\in k-1$, $|u_{i}|\leq m_{j}<|w|$, then $v(m_{j})=0$. For the following, it is important to notice that $q\geq k-2$. If for some $j<j^{\prime}$, $|c_{m_{j}}|<|u_{i}|$ and $|w|\leq|c_{m_{j^{\prime}}}|$, then $c_{m_{j^{\prime}}}(m_{j})=0$, contrary to our assumption that $\{c_{m_{j}}:j\in k-1\}$ codes a $(k-1)$-clique. Lastly, suppose $m_{0}\geq|w|$. Then the nodes $c_{m_{j}}$, $j\in k-1$, are exactly the coding nodes $c_{n-k+1},\dots,c_{n-1}$. Thus, $v(c_{m_{0}})=0$. Therefore, by the $k$-FBC, $v$ is in $\mathbb{S}_{k}(n+1)$. Let $c_{n}=v$ and split according to the $k$-FBC to construct $\mathbb{S}_{k}(n+2)$. This satisfies (1)–(4) and (i) and (ii).

This inductive process constructs a tree $\mathbb{S}_{k}=\bigcup_{n<\omega}\mathbb{S}_{k}(n)$ which is a strong $K_{k}$-free tree satisfying (i) and (ii).

Remark 4.19*.*

For $k=3$, the previous construction of a strong triangle-free tree produces a strong triangle-free tree in the sense of [14], albeit in a more streamlined fashion.

Example 4.20** (A Strong $K_{3}$-Free Tree, $\mathbb{S}_{3}$).**

In keeping with the construction method above, we present the first several levels of the construction of $\mathbb{S}_{3}$. Let $u_{0}$ denote the empty sequence, and suppose $u_{1}=\langle 1\rangle$ and $u_{2}=\langle 0\rangle$. The ghost coding node is $c_{-1}=\langle\rangle$, the empty sequence. The coding nodes $c_{n}$ where $n$ is odd will be $c_{n}={0^{(n)}}^{\frown}1$. In particular, $c_{1}=\langle 0,1\rangle$, $c_{3}=\langle 0,0,0,1\rangle$, $c_{5}=\langle 0,0,0,0,0,1\rangle$, etc. These nodes are in every tree satisfying the $K_{3}$-Free Branching Criterion.

Let $c_{0}=\langle 1\rangle$; this node extends $u_{0}=\langle\rangle$. Let $c_{2}=\langle 1,0,1\rangle$; this node extends $u_{1}=\langle 1\rangle$. Let $c_{4}=\langle 0,1,0,0,1\rangle$; this node extends $u_{2}=\langle 0\rangle$. If $u_{3}=\langle 1,1\rangle$, since this node is not in $\mathbb{S}_{3}(7)$, we let $c_{6}={0^{(6)}}^{\frown}1$. If $u_{3}=\langle 1,0\rangle$, we can let $c_{6}$ be any node in $\mathbb{S}_{3}(7)$ extending $u_{3}$. For instance, if we care about making $\mathbb{S}_{3}$ recursively definable with respect to the sequence $\langle u_{i}:i\in\mathbb{N}\rangle$, we can let $c_{6}$ be the rightmost extension of $u_{3}$ in $\mathbb{S}_{3}(7)$ which has last entry $1$, namely $c_{6}=\langle 1,0,1,0,1,0,1\rangle$. In this manner, one constructs a tree such as in Figure 3.

Example 4.21** (A Strong $K_{4}$-Free Tree, $\mathbb{S}_{4}$).**

The following tree $\mathbb{S}_{4}$ in Figure 4. is an example of a strong $K_{4}$-free tree. Let $u_{0}$ denote the empty sequence, and suppose $u_{1}=\langle 1\rangle$ and $u_{2}=\langle 0\rangle$. The ghost coding nodes are $c_{-2}=\langle\rangle$ and $c_{-1}=\langle 1\rangle$. According to the construction in Example 4.18, the first three coding nodes of $\mathbb{S}_{4}$ are $c_{0}=\langle 1,1\rangle$, which extends $u_{0}$, $c_{1}=\langle 0,1,1\rangle$, and $c_{2}=\langle 0,0,1,1\rangle$, each time splitting according to the $K_{4}$-Free Splitting Criterion ($4$-FSC) to construct a tree $r_{5}(\mathbb{S}_{4})$. Split again according to the $4$-FSC to construct the next level of the tree, $\mathbb{S}_{4}(5)$. Since $u_{1}$ is in $r_{3}(\mathbb{S}_{4})$, letting $c_{3}=\langle 1,1,0,1,1\rangle$ satisfies requirement (i) in Example 4.18. Let $c_{4}=\langle 0,0,0,0,1,1\rangle$ and $c_{5}=\langle 0,0,0,0,0,1,1\rangle$. Since $u_{2}=\langle 0\rangle$ is in $r_{4}(\mathbb{S}_{4})$, taking $c_{6}$ to be $\langle 0,1,1,0,0,0,1,1\rangle$ satisfies our requiremnts. One can check that this node is in $\mathbb{S}_{4}(8)$. (One could also simply let $c_{6}$ be ${0^{(6)}}^{\frown}\langle 1,1\rangle$. Continue the construction according to Example 4.18.

As in the case of $\mathcal{H}_{3}$ in [14], the purpose of not allowing coding nodes to split is to reduce the number of different types of trees coding a given finite $K_{k}$-free graph. Having the coding nodes be dense in the tree is necessary development of Ramsey theory. However, the same example of a bad coloring as given in Example 3.18 of [14] provides a bad coloring for any strong $K_{k}$-free tree, for any $k\geq 3$.

Example 4.22** (A bad coloring of vertices in $\mathcal{H}_{k}$).**

Let $k\geq 3$ be fixed. Color all coding nodes in $\mathbb{S}_{k}$ extending $\langle 1\rangle$ red. In particular, $c_{0}=\langle 1\rangle$ is colored red. Given $n\geq 0$, suppose that for each $i\leq n$, all coding nodes in $\mathbb{S}_{k}$ extending ${0^{(i)}}^{\frown}1$ have been colored either red or blue. Look at the coding node $c_{n}$. This node has length $n+1$ and has already been assigned a color. If $c_{n}$ is red, then color every coding node extending ${0^{(n+1)}}^{\frown}1$ blue; if $c_{n}$ is blue, then color every coding node extending ${0^{(n+1)}}^{\frown}1$ red. Notice that any subtree of $\mathbb{S}_{k}$ which is strongly similar to $\mathbb{S}_{k}$ in the sense of Definition 3.7 where additionally coding nodes are sent to coding nodes, has nodes of both colors. (See Definition 5.3 for the precise definition of strongly similar for trees with coding nodes.) Equivalently, any subtree of $\mathbb{S}_{k}$ which is again a strong $K_{k}$-tree which represents a copy of $\mathcal{H}_{k}$ has coding nodes with both colors.

This coloring in Example 4.22 is equivalent to a coloring of the vertices of $\mathcal{H}_{k}$. Recall that the work of Komjáth and Rödl in [35] for $k=3$ and work of El-Zahar and Sauer in [20] for $k\geq 4$, shows that for any coloring of the vertices in $\mathcal{H}_{k}$ into two colors, there is a subgraph $\mathcal{H}^{\prime}$ which is again a $K_{k}$-free Henson graph in which all vertices have the same color. However, the previous example shows that we cannot expect the subgraph $\mathcal{H}^{\prime}$ to have induced tree with coding nodes $T_{\mathcal{H}^{\prime}}$ strongly similar to $\mathbb{S}_{k}$. Since we are aiming to prove Ramsey theorems on collections of trees with coding nodes which are all strongly similar to each other, we immediately turn to the next section where we present the skewed version of these trees on which the relevant Ramsey theory can be developed.

5. Strong $\mathcal{H}_{k}$-coding trees

The classes of trees coding Henson graphs on which we develop Ramsey theory are presented in this section. For each $k\geq 3$, fixing a tree $\mathbb{S}_{k}$ constructed as in Example 4.18, we construct its skew version, denoted $\mathbb{T}_{k}$, the skewing being necessary to avoid the bad colorings seen in Example 4.22. The coding nodes in $\mathbb{T}_{k}$ code a $k$-clique-free Henson graph in the same way as the coding nodes of $\mathbb{S}_{k}$. In Definition 5.12, we present the space of strong $\mathcal{H}_{k}$-coding subtrees of $\mathbb{T}_{k}$. These are subtrees of $\mathbb{T}_{k}$ which are isomorphic to $\mathbb{T}_{k}$ in a strong way, and consequently code a copy of $\mathcal{H}_{k}$ in the same way that $\mathbb{T}_{k}$ does.

By the end of Section 8, these spaces of strong $\mathcal{H}_{k}$-coding trees will be shown to have Ramsey theorems similar to the Milliken space of strong trees [41]. The added difficulty for $k>3$ will be seen and addressed from here throughout the rest of the paper. This section extends results of Section 4 in [14] to $\mathcal{H}_{k}$ for all $k\geq 4$, while providing a new, more streamlined approach for the $k=3$ case.

5.1. Definitions, notation, and maximal strong $\mathcal{H}_{k}$-coding trees, $\mathbb{T}_{k}$

The following terminology and notation will be used throughout, some of which is recalled from previous sections for ease of reading. A subset $X\subseteq\mathcal{S}$ is a level set if all nodes in $X$ have the same length. We continue to use the notions of tree and tree with coding nodes given in Definitions 3.3 and 4.1, respectively, augmented to include ghost coding nodes, as was the case in the definition of $\mathbb{S}_{k}$ in Example 4.18. Recall from equation (5) (just after Definition 3.3) that for a tree $T$, we define $\widehat{T}$ to be the tree of all nodes in $s\in\mathcal{S}$ for which there is some $t\in T$ such that $s\subseteq t$.

Let $T\subseteq\mathcal{S}$ be a finite or infinite tree with coding nodes $\langle c^{T}_{n}:n\in N\rangle$, where either $N\in\mathbb{N}$ or $N=\mathbb{N}$. If $T$ is to be a strong $\mathcal{H}_{k}$-coding tree, then $T$ will also have ghost coding nodes $\langle c^{T}_{-k+2},\dots,c^{T}_{-1}\rangle$. We let $l^{T}_{n}$ denote $|c^{T}_{n}|$, the length of $c^{T}_{n}$. Recall that $|c^{T}_{n}|$ is the domain of $c^{T}_{n}$, as the sequence $c^{T}_{n}$ is a function from some natural number into $\{0,1\}$. We sometimes drop the superscript $T$ when it is clear from the context. A node $s\in T$ is called a splitting node if both $s^{\frown}0$ and $s^{\frown}1$ are in $\widehat{T}$; equivalently, $s$ is a splitting node in $T$ if there are nodes $s_{0},s_{1}\in T$ such that $s_{0}\supseteq s^{\frown}0$ and $s_{1}\supseteq s^{\frown}1$. The critical nodes of $T$ is the set of all splitting and coding nodes, as well as any ghost coding nodes of $T$. Given $t$ in $T$, let $T\restriction|t|$ denote the set of all $s\in T$ such that $|s|=|t|$, and call $T\restriction|t|$ a level of $T$.

We will say that a tree $T$ is skew if each level of $T$ has exactly one of either a coding node or a splitting node. The set of levels of a skew tree $T\subseteq\mathcal{S}$, denoted $L(T)$, is the set of those $l\in\mathbb{N}$ such that $T$ has either a splitting or a coding node of length $l$. A skew tree $T$ is strongly skew if additionally for each splitting node $s\in T$, every $t\in T$ such that $|t|>|s|$ and $t\not\supset s$ also satisfies $t(|s|)=0$; that is, the passing number of any node passing by, but not extending, a splitting node is [math].

Given a skew tree $T$, we let $\langle d^{T}_{m}:m\in M\rangle$ denote the enumeration of all critical nodes of $T$ in increasing order; $d^{T}_{0}$ will be the stem of $T$, that is, the first splitting node of $T$. Appropriating the standard notation for Milliken’s strong trees, for each $m\in M$, the $m$-th level of $T$ is

[TABLE]

Then for any skew tree $T$,

[TABLE]

For $m\in M$, the $m$-th approximation of $T$ is defined to be

[TABLE]

Let $m_{n}$ denote the integer such that $c^{T}_{n}\in T(m_{n})$; thus, $d^{T}_{m_{n}}=c^{T}_{n}$.

For $n\geq 1$, the $n$-th interval of $T$ is $\bigcup\{T(j):m_{n-1}<j\leq m_{n}\}$; the interval of $T$ between $c^{T}_{n-1}$ and $c^{T}_{n}$ is $\bigcup\{T(j):m_{n-1}<j<m_{n}\}$. The [math]-th interval of $T$ is $\bigcup\{T(j):j\leq m_{0}\}$. We call a skew tree $T$ regular if for each $n\in N$, the lengths of the splitting nodes in the $n$-th interval of $T$ increase as their lexicographic order decreases.

In contrast to our approach in [14] where we defined strong $\mathcal{H}_{3}$-coding trees via several structural properties, in this paper we shall construct a particular strong $\mathcal{H}_{k}$-coding tree $\mathbb{T}_{k}$ and then define a subtree to be a strong $\mathcal{H}_{k}$-coding tree if it is isomorphic to $\mathbb{T}_{k}$ in a strong sense, to be made precise in Definition 5.12. The coding structure of $\mathbb{T}_{k}$ is the same as that of the strong $K_{k}$-free tree $\mathbb{S}_{k}$ given in Example 4.18. The best way to think about $\mathbb{T}_{k}$ is that it is simply the strongly skew, regular version of $\mathbb{S}_{k}$.

Example 5.1** (Construction Method for Maximal Strong $\mathcal{H}_{k}$-Coding Trees, $\mathbb{T}_{k}$).**

Fix $k\geq 3$, and let $\langle u_{i}:i\in\mathbb{N}\rangle$ be any enumeration of the nodes in $\mathcal{S}$ such that $|u_{i}|\leq|u_{j}|$ whenever $i<j$. Let $\mathbb{S}_{k}$ be a strong $K_{k}$-free tree constructed in Example 4.18. Recall that the graph represented by the coding nodes $\langle c^{\mathbb{S}_{k}}_{n}:n\in\mathbb{N}\rangle$ in $\mathbb{S}_{k}$ is the $k$-clique-free Henson graph. (The graph represented by the coding nodes in $\mathbb{S}_{k}$ along with the ghost coding nodes in $\mathbb{S}_{k}$ is also a $k$-clique-free Henson graph, since $(A)^{\tt tree}_{k}$ holds.) Define $\mathbb{T}_{k}$ to be the tree obtained by stretching $\mathbb{S}_{k}$ to make it strongly skew and regular while preserving the passing numbers so that $\mathbb{T}_{k}$ represents the same copy of the Henson graph $\mathcal{H}_{k}$ as $\mathbb{S}_{k}$ does. $\mathbb{T}_{k}$ will have coding nodes $\langle c^{k}_{n}:n\in\mathbb{N}\rangle$ and ghost coding nodes $\langle c^{k}_{-k+2},\dots,c^{k}_{-1}\rangle$. For all pairs $j<n$, we will have $c^{k}_{n}(|c^{k}_{j}|)=c^{\mathbb{S}_{k}}_{n}(|c^{\mathbb{S}_{k}}_{j}|)$. The $m$-th critical node $d^{k}_{m}$ in $\mathbb{T}_{k}$ will be a node of length $m$, so that $\mathbb{T}_{k}$ will have nodes in every length in $\mathbb{N}$. The critical nodes consist of the splitting nodes, coding nodes, and ghost coding nodes. In particular, we will have $\widehat{\mathbb{T}}_{k}=\mathbb{T}_{k}$, and $\mathbb{T}_{k}(m)=\mathbb{T}_{k}\restriction m$ for each $m\in\mathbb{N}$.

We now show precisely how to construct $\mathbb{T}_{k}$, given $\mathbb{S}_{k}$. Set $m_{-k+2}=0$ and $\mathbb{T}_{k}(0)=\mathbb{S}_{k}(0)$, which is the singleton $\{\langle\rangle\}$. Let the ghost coding node of $\mathbb{T}_{k}$ be $c^{k}_{-k+2}=\langle\rangle$. This node splits so that $\mathbb{T}_{k}(1)=\mathbb{S}_{k}(1)$, which has exactly two nodes, so there is a bijection between these level sets of nodes. As in the previous section, let $r_{m}(\mathbb{T}_{k})$ denote $\bigcup_{j<m}\mathbb{T}_{k}(j)$. Then $r_{2}(\mathbb{T}_{k})=\mathbb{T}_{k}(0)\cup\mathbb{T}_{k}(1)$. The [math]-th critical node $d^{k}_{0}$ equals the ghost coding node $c_{-k+2}^{k}$.

Suppose that $n\geq-k+2$ and that $r_{m_{n}+2}(\mathbb{T}_{k})$ has been constructed, $c^{k}_{n}\in\mathbb{T}_{k}(m_{n})$ has been fixed, and there is a bijection between $\mathbb{T}_{k}(m_{n}+1)$ and $\mathbb{S}_{k}(n+k-1)$. Let $J$ be the number of nodes in $\mathbb{S}_{k}(n+k-1)$ which split into two extensions in $\mathbb{S}_{k}(n+k)$. Let $m_{n+1}=m_{n}+J+1$. (Notice that for $k=3$, $m_{0}=2$, and for $k=4$, $m_{-1}=3$; more generally, for $k\geq 4$, $m_{-k+3}=3$.) Let $\{\tilde{s}_{j}:m_{n}<j<m_{n+1}\}$ enumerate in reverse lexicographic order those members of $\mathbb{S}_{k}(n+k-1)$ which split into two extensions in $\mathbb{S}_{k}(n+k)$. Let $\varphi$ be the lexicographic preserving bijection from $\mathbb{S}_{k}(n+k-1)$ onto $\mathbb{T}_{k}(m_{n}+1)$. For each $m_{n}<j<m_{n+1}$, define $s_{j}=\varphi(\tilde{s}_{j})$, and set

[TABLE]

where $i_{j}=j-m_{n}-1$. Thus, $d^{k}_{m_{n}+1}=s_{m_{n}+1}$, $d^{k}_{m_{n}+2}={s_{m_{n}+2}}^{\frown}\langle 0\rangle$, $d^{k}_{m_{n}+3}={s_{m_{n}+2}}^{\frown}\langle 0,0\rangle$, and finally, $d^{k}_{m_{n+1}-1}={s_{m_{n+1}-1}}^{\frown}0^{(J-1)}$. These are the splitting nodes in the interval of $\mathbb{T}_{k}$ between $c^{k}_{n}$ and $c^{k}_{n+1}$. For each $i\in\{0,1\}$, define $t_{j}^{i}$ to be the binary sequence of length $m_{n+1}$ which extends ${d^{k}_{j}}^{\frown}i$ by all [math]’s. Define

[TABLE]

where $S_{n+1}^{\prime}=\mathbb{S}_{k}(n+k)\setminus\{\tilde{s}_{j}:m_{n}<j<m_{n+1}\}$ and for each $\tilde{s}\in S_{n+1}^{\prime}$, $t_{\tilde{s}}$ is the extension of $\varphi(\tilde{s})$ by [math]’s to length $m_{n+1}$. Let $c^{k}_{n+1}$ be the leftmost extension in $\mathbb{T}_{k}(m_{n+1})$ of $\varphi(\tilde{s}_{c})$, where $\tilde{s}_{c}$ is the leftmost immediate successor of $c_{n+1}^{\mathbb{S}_{k}}$ in $\mathbb{S}_{k}(n+k-1)$. Define

[TABLE]

Notice that the level sets $\mathbb{S}_{k}(n+k)$, $\mathbb{T}_{k}(m_{n+1})$, and $\mathbb{T}_{k}(m_{n+1}+1)$ have the same cardinality. Moreover, the lexicographic preserving bijection from $\mathbb{S}_{k}(n+k)$ onto $\mathbb{T}_{k}(m_{n+1})$ preserves passing numbers. By our construction, for each $m\in\mathbb{N}$, $|d^{k}_{m}|=m$.

Let $\mathbb{T}_{k}=\bigcup_{m\in\mathbb{N}}\mathbb{T}_{k}(m)$ with coding nodes $\langle c^{k}_{n}:n\in\mathbb{N}\rangle$ and ghost coding nodes $\langle c^{k}_{-k+2},\dots,c^{k}_{-1}\rangle$. $\mathbb{T}_{k}$ is a strongly skew, regular tree with coding nodes representing a copy of $\mathcal{H}_{k}$ and is maximal in the sense that $\widehat{\mathbb{T}}_{k}=\mathbb{T}_{k}$

Remark 5.2*.*

Notice that the coding nodes (along with the ghost coding nodes) in $\mathbb{T}_{k}$ represent the same ordered copy of $\mathcal{H}_{k}$ as the one represented by the coding nodes (along with the ghost coding nodes) in $\mathbb{S}_{k}$. That is, given $j<n$, $c^{k}_{n}(|c^{k}_{j}|)=c^{\mathbb{S}_{k}}_{n}(|c^{\mathbb{S}_{k}}_{j}|)$. A simple way to think about $\mathbb{T}_{k}$ is that it is the skew tree such that if one “zips up” the splitting nodes in the $n$-th interval of $\mathbb{T}_{k}$ to the length of $c^{k}_{n}$, then one recovers a tree which is strongly similar (in fact, strongly isomorphic–see Definition 5.10) to $\mathbb{S}_{k}$, in the sense of the upcoming Definition 5.3.

5.2. The space of strong $\mathcal{H}_{k}$-coding trees

Let $k\geq 3$ be given, and fix $\mathbb{T}_{k}$ as constructed as in Example 5.1. In preparation for defining the space of strong $\mathcal{H}_{k}$-coding subtrees of $\mathbb{T}_{k}$, we provide the following definitions.

A subset $A$ of $\mathbb{T}_{k}$ is an antichain if for all $s,t\in A$, $s\subseteq t$ implies $s=t$. Recall that a subset $X$ of $\mathbb{T}_{k}$ is a level set if all members of $X$ have the same length. Thus, each level set is an antichain. Given a subset $S\subseteq\mathbb{T}_{k}$, recall that the meet closure of $S$, denoted $S^{\wedge}$, is the set of all meets of (not necessarily distinct) pairs of nodes in $S$; in particular, $S^{\wedge}$ contains $S$. We say that $S$ is meet-closed if $S=S^{\wedge}$. Recall that for $s\in\mathcal{S}$ and $l\in\mathbb{N}$ with $|s|\geq l$, $s\restriction l$ denotes the initial segment of $s$ of length $l$. Similarly to equation (8), given a subset $S\subseteq\mathbb{T}_{k}$ and any $l\in\mathbb{N}$, we define

[TABLE]

That is, $S\restriction l$ is the set of nodes in $\widehat{S}$ of length exactly $l$ which are initial segments of some node in $S$. Thus, $S\restriction l=\widehat{S}\restriction l$, whether or not $S$ has nodes of length $l$.

If $s\in\mathbb{T}_{k}$ has the same length as some (ghost) coding node $c^{k}_{n}$ in $\mathbb{T}_{k}$, then $s$ has a unique immediate successor in $\mathbb{T}_{k}$; denote this as $s^{+}$. We say that $s$ has passing number $i$ at $c^{k}_{n}$ exactly when $s^{+}(|c^{k}_{n}|)=i$.

The following is Definition 4.9 in [14] augmented to include ghost coding nodes; it extends Definition 3.7 of Sauer. It is important to note that a ghost coding node in some subset $T\subseteq\mathbb{T}_{k}$ is never a coding node in $\mathbb{T}_{k}$.

Definition 5.3** (Strong Similarity Map).**

Let $k\geq 3$ be fixed, and let $S,T\subseteq\mathbb{T}_{k}$ be meet-closed subsets. A function $f:S\rightarrow T$ is a strong similarity map of $S$ to $T$ if for all nodes $s,t,u,v\in S$, the following hold:

1. (1)

$f$ is a bijection. 2. (2)

$f$ preserves lexicographic order: $s<_{\mathrm{lex}}t$ if and only if $f(s)<_{\mathrm{lex}}f(t)$. 3. (3)

$f$ preserves meets, and hence splitting nodes: $f(s\wedge t)=f(s)\wedge f(t)$. 4. (4)

$f$ preserves relative lengths: $|s\wedge t|<|u\wedge v|$ if and only if $|f(s)\wedge f(t)|<|f(u)\wedge f(v)|$. 5. (5)

$f$ preserves initial segments: $s\wedge t\subseteq u\wedge v$ if and only if $f(s)\wedge f(t)\subseteq f(u)\wedge f(v)$. 6. (6)

$f$ preserves (ghost) coding nodes: $s$ is a coding node in $S$ if and only if $f(s)$ is a coding node in $T$. If $S$ has also ghost coding nodes, then $s$ is a ghost coding node in $S$ if and only if $f(s)$ is a ghost coding node in $T$. 7. (7)

$f$ preserves passing numbers at (ghost) coding nodes: If $c$ is a (ghost) coding node in $S$ and $s$ is a node in $S$ with $|s|=|c|$, then $(f(s))^{+}(|f(c)|)=s^{+}(|c|)$. In words, the passing number of $f(s)$ at $f(c)$ equals the passing number of $s$ at $c$.

We say that $S$ and $T$ are strongly similar, and write $S\stackrel{{\scriptstyle s}}{{\sim}}T$, exactly when there is a strong similarity map between $S$ and $T$.

It follows from (3) that $s\in S$ is a splitting node in $S$ if and only if $f(s)$ is a splitting node in $T$. In all cases above, it may be that $s=t$ and $u=v$, so in particular, (5) implies that $s\subseteq u$ if and only if $f(s)\subseteq f(u)$. Notice that strong similarity is an equivalence relation, since the inverse of a strong similarity map is a strong similarity map, and the composition of two strong similarity maps is a strong similarity map. If $T^{\prime}\subseteq T$ and $f$ is a strong similarity of $S$ to $T^{\prime}$, then we say that $f$ is a strong similarity embedding of $S$ into $T$.

Our goal in this subsection is to define a space of subtrees of $\mathbb{T}_{k}$ for which the development of a Ramsey theory akin to the Halpern-Läuchli and Milliken Theorems is possible. The first potential obstacle arises due to the fact that $k$-cliques are forbidden in $\mathcal{H}_{k}$. This manifests in terms of trees in the following way: There are finite subtrees of $\mathbb{T}_{k}$ which are strongly similar to an initial segment of $\mathbb{T}_{k}$, and yet cannot be extended within $\mathbb{T}_{k}$ to a subtree which is strongly similar to $\mathbb{T}_{k}$. In this subsection we make precise what the possible obstructions are. We then define the set of strong $\mathcal{H}_{k}$-coding trees to be those trees which avoid the possible obstructions.

The next definitions are new to $k$-clique-free graphs for $k>3$, and are necessary for the work in this paper. When $k=3$, the rest of this section simply reproduces the concepts of sets of parallel $1$’s and the Parallel $1$’s Criterion used throughout [14], though in a new and more streamlined manner. Fix $k\geq 3$ throughout the rest of Section 5.

If $X$ is a level subset of some meet-closed $S\subseteq\mathbb{T}_{k}$, let $l_{X}$ denote the length of the members of $X$. If the nodes in $X$ are not maximal in $S$, let the set of immediate successors of $X$ in $\widehat{S}$ be denoted by $X^{+}$. Thus, when $S$ is understood,

[TABLE]

Note that if $l_{X}$ is the length of a (ghost) coding node in $\mathbb{T}_{k}$, then each node in $X$ has a unique extension in $X^{+}$ which is determined by $\mathbb{T}_{k}$, irregardless of $S$.

Definition 5.4** (Pre-$a$-Clique and Witnesses).**

Let $a\in[3,k]$, and let $X\subseteq\mathbb{T}_{k}$ be a level subset of $\mathbb{T}_{k}$. ($X$ is allowed to consist of a single node.) Let $l_{X}$ denote the length of the nodes in $X$. We say that $X$ has pre-$a$-clique if there is an index set $I\subseteq[-k+2,-1]\cup\mathbb{N}$ of size $a-2$ such that, letting $i_{*}=\max(I)$ and $l=|c^{k}_{i_{*}}|$, the following hold:

1. (1)

The set $\{c^{k}_{i}:i\in I\}$ codes an $(a-2)$-clique; that is, for each pair $i<j$ in $I$, $c^{k}_{j}(|c^{k}_{i}|)=1$; 2. (2)

For each node $x\in X$ and each $i\in I$, $x^{+}(|c^{k}_{i}|)=1$.

We say that $X$ has a pre-$a$-clique at $l$, and that $X\restriction l$ is a pre-$a$-clique. The set of nodes $\{c^{k}_{i}:i\in I\}$ is said to witness that $X$ has a pre-$a$-clique at $l$.

Notation 5.5**.**

We write $\mbox{P}_{a}(X)$ exactly when $X$ has a pre-$a$-clique.

Remark 5.6*.*

Whenever a level set $X$ has a pre-$k$-clique, then for any set $C$ of coding (and/or ghost coding) nodes of $\mathbb{T}_{k}$ witnessing that $\mbox{P}_{k}(X)$, the set $C$ codes a $(k-2)$-clique, and each $x\in X$ has passing number $1$ at each $c\in C$. It follows that, if $X$ has more than one node, then for any coding node $c$ in $\mathbb{T}_{k}$ extending some $x\in X$, any extension of any $y\in X\setminus\{x\}$ to some node $y^{\prime}$ of length greater than $|c|$ must satisfy $y^{\prime}(|c|)=0$.

Thus, the nodes in a pre-$k$-clique are ‘entangled’: The splitting possibilities in the cone above one of these nodes depends on the cones above the other nodes in the pre-$k$-clique. If $X$ is contained in some finite subtree $A$ of $\mathbb{T}_{k}$ and $\mbox{P}_{k}(X)$ is not witnessed by coding nodes in $A$, then the graph coded by $A$ has no knowledge that the cones above $X$ in $\mathbb{T}_{k}$ are entangled. Then no extension of $A$ into $\mathbb{T}_{k}$ can be strongly similar to $\mathbb{T}_{k}$. This is one of the main reasons that developing Ramsey theory for Henson graphs is more complex than for the Rado graph.

Even if $a<k$, pre-$a$-cliques are also entangled. In the set-up to the space of strong coding trees, we must consider pre-$a$-cliques for each $a\in[3,k]$; it is necessary to witness them in order to guarantee the existence of extensions within a given strong $\mathcal{H}_{k}$-coding tree $T$ which are strongly similar to some tree which should exist according to the $K_{k}$-Free Criterion. The guarantee of such extensions is the heart of the extension lemmas in Section 6.

Definition 5.7** (New Pre-$a$-Clique).**

Let $a\in[3,k]$, and let $X\subseteq\mathbb{T}_{k}$ be a level set. ($X$ can consist of a single coding node.) We say that $X$ has a new pre-$a$-clique at $l$ if $X\restriction l$ is a pre-$a$-clique and for each $l^{\prime}<l$ for which $X\restriction l$ and $X\restriction l^{\prime}$ have the same number of nodes, $X\restriction l^{\prime}$ is not a pre-$a$-clique.

The reasoning behind the requirement that $X\restriction l$ and $X\restriction l^{\prime}$ have the same number of nodes will become more apparent in latter sections, when we want to color finite antichains of coding nodes coding a copy of some finite $K_{k}$-free graph.

Definition 5.8**.**

Let $a\in[3,k]$, and let $X,Y\subseteq\mathbb{T}_{k}$ be level sets with $l_{Y}>l_{X}$. We say that $Y$ has no new pre-$a$-cliques over $X$ if and only if the following holds: For each $j\in(l_{X},l_{Y}]$ and each $Z\subseteq Y$, if $Z\restriction j$ is a pre-$a$-clique, then $Z$ end-extends $Z\restriction l_{X}$, and $Z\restriction l_{X}$ already has a pre-$a$-clique. We say that $Y$ has no new pre-cliques over $X$ if $Y$ has no new pre-$a$-cliques over $X$ for any $a\in[3,k]$.

For example, suppose $X$ is a singleton which is a new pre-$a$-clique for some $a\in[3,k]$ and $Z$ is a level set with at least two members such that $Z\restriction l_{X}=X$. If for some $l_{X}<j\leq l_{Z}$, $Z\restriction j$ has at least two distinct nodes and for each $z\in Z$, $z(j)=1$, then $Z$ has at least a new pre-$3$-clique over $X$.

The next definition gives precise conditions under which a new pre-$a$-clique at $l$ in a subtree $T$ of $\mathbb{T}_{k}$ is maximal in the interval of $T$ containing $l$.

Definition 5.9** (Maximal New Pre-$a$-Clique).**

Let $T$ be a subtree of $\mathbb{T}_{k}$, let $X\subseteq T$ be a level set, and let $a\in[3,k]$. We say that $X$ has a maximal new pre-$a$-clique in $T$ at $l$ if $X\restriction l$ is a new pre-$a$-clique in $T$ which is also maximal in $T$ in the following sense: Let $d$ denote the critical node in $T$ of maximum length satisfying $|d|<l$. If $m$ is the index so that $d=d^{T}_{m}$, let $e$ denote $d^{T}_{m+1}$ and note that $l\leq|e|$. Then for any $l^{\prime}\in(l,|e|]$ and any new pre-$a$-clique $Y\subseteq T\restriction l^{\prime}$, if $Y\restriction l$ contains $X\restriction l$ then these sets are equal; hence $l^{\prime}=l$, since $T$ has no splitting nodes in the interval $(|d|,|e|)$.

We write $\mbox{MP}_{a}(X;T)$ if $X$ has a maximal new pre-$a$-clique in $T$ in the interval of $T$ containing the length of the nodes in $X$. Thus, if $l_{X}=|d^{T}_{m}|$, then $\mbox{MP}_{a}(X;T)$ means that for some $l\in(l^{T}_{m-1},l^{T}_{m}]$, $X$ has a maximal new pre-$a$-clique at $l$.

In Definition 5.9, for any level set $Z$ end-extending $X$, we say that $Z$ has a maximal new pre-$a$-clique in $T$ at $l$. We will say that a set $Y\subseteq T$ contains a maximal new pre-$a$-clique at $l$ if $\mbox{MP}_{a}(X;T)$ for some subset $X\subseteq Y\restriction l$.

Definition 5.10** (Strong Isomorphism).**

Let $S$ and $T$ be strongly similar subtrees of $\mathbb{T}_{k}$ with $M$ many critical nodes. The strong similarity map $f:T\rightarrow S$ is a strong isomorphism if for each positive $m\in M$, the following hold: For each $a\in[3,k]$, a level subset $X\subseteq T\restriction|d^{T}_{m}|$ has a maximal new pre-$a$-clique in $T$ in the interval $(|d^{T}_{m-1}|,|d^{T}_{m}|]$ if and only if $f[X]$ has a maximal new pre-$a$-clique in $S$ in the interval $(|d^{S}_{m-1}|,|d^{S}_{m}|]$.

When there is a strong isomorphism between $S$ and $T$, we say that $S$ and $T$ are strongly isomorphic and write $S\cong T$.

Observation 5.11**.**

The relation $\cong$ is an equivalence relation, since the inverse of a strong isomorphism is a strong isomorphism, and composition of two strong isomorphisms is again a strong isomorphism. Moreover, since strong similarity maps preserve (ghost) coding nodes and passing numbers, any strong isomorphism $f:T\rightarrow S$ will map a set of coding (and/or ghost coding) nodes in $T$ witnessing a pre-$a$-clique $X$ in $T$ to a set of coding (and/or ghost coding) nodes in $S$ witnessing that $f[X]$ is a pre-$a$-clique $X$ in $S$.

Strong isomorphisms preserve all relevant structure regarding the shape of the tree, (ghost) coding nodes and passing numbers, and maximal new pre-$a$-cliques and their witnesses. They provide the essential structure of our strong $\mathcal{H}_{k}$-coding trees.

Definition 5.12** (The Space of Strong $\mathcal{H}_{k}$-Coding Trees $(\mathcal{T}_{k},\leq,r)$).**

A tree $T\subseteq\mathbb{T}_{k}$ (with designated ghost coding nodes) is a member of $\mathcal{T}_{k}$ if and only if there is a strong isomorphism, denoted $f_{T}$, from $\mathbb{T}_{k}$ onto $T$. Thus, $\mathcal{T}_{k}$ consists of all subtrees of $\mathbb{T}_{k}$ which are strongly isomorphic to $\mathbb{T}_{k}$. Call the members of $\mathcal{T}_{k}$ strong $\mathcal{H}_{k}$-coding trees, or just strong coding trees when $k$ is clear. The partial ordering $\leq$ on $\mathcal{T}_{k}$ is simply inclusion: For $S,T\in\mathcal{T}_{k}$, we write $S\leq T$ if and only if $S$ is a subtree of $T$. If $T\in\mathcal{T}_{k}$, the notation $S\leq T$ implies that $S$ is also a member of $\mathcal{T}_{k}$.

Given $T\in\mathcal{T}_{k}$, for $n\in[-k+2,-1]\cup\mathbb{N}$, we let $c^{T}_{n}$ denote $f_{T}(c^{k}_{n})$; hence, $\langle c^{T}_{-k+2},\dots,c^{T}_{-1}\rangle$ enumerates the ghost coding nodes and $\langle c^{T}_{n}:n\in\mathbb{N}\rangle$ enumerates the coding nodes in $T$ in increasing order of length. Letting $\langle d^{T}_{m}:m\in\mathbb{N}\rangle$ enumerate the critical nodes of $T$ in order of increasing length, note that the $m$-th critical node in $T$, $d^{T}_{m}$, equals $f_{T}(d^{k}_{m})$. In particular, the ghost coding node $c^{T}_{-k+2}$ equals $d^{T}_{0}$, the minimum splitting node in $T$.

It follows from the definition of strong isomorphism and the structure of $\mathbb{T}_{k}$ that each coding node in $T$ is a singleton pre-k-clique, while each ghost coding node is not. Precisely, $c_{-k+2}^{T}$ is not a pre-clique at all. If $k\geq 4$, then $c_{-k+3}^{T}$ is a pre-$3$-clique, and in general, for $3\leq i<k$, the ghost coding node $c^{T}_{-k+i}$ is a pre-$i$-clique. Thus, the ghost coding nodes in $T$ cannot be coding nodes in $\mathbb{T}_{k}$; hence the name. It is the case, though, that each ghost coding node $c^{T}_{n}$ with $n>-k+2$ in $T$ has the same length as some ghost or coding node in $\mathbb{T}_{k}$.

Let $T(m)$ denote the set of nodes in $T$ of length $|d^{T}_{m}|$. Recalling the notation in (28) and (30) in Subsection 5.1, the $m$-th finite approximation of $T$ is

[TABLE]

for $m\in\mathbb{N}$. This is equal to $\{t\in T:|t|<|d^{T}_{m}|\}$, since $T$ is a tree. Thus for $m<n$, $r_{n}(T)$ end-extends $r_{m}(T)$, and $T=\bigcup_{m\in\mathbb{N}}r_{m}(T)$. Notice that for any tree $T$, $r_{0}(T)$ is the emptyset and $r_{1}(T)=\{d^{T}_{0}\}$, the splitting node of smallest length in $T$.

For each $m\in\mathbb{N}$, define

[TABLE]

and let

[TABLE]

Given $A\in\mathcal{AT}^{k}$ and $T\in\mathcal{T}_{k}$, define

[TABLE]

Given $j<m$, $A\in\mathcal{AT}^{k}_{j}$ and $T\in\mathcal{T}_{k}$, define

[TABLE]

For $A\in\mathcal{AT}^{k}$ and $B\in\mathcal{AT}^{k}\cup\mathcal{T}_{k}$, if for some $m$, $r_{m}(B)=A$, then we write $A\sqsubseteq B$ and say that $A$ is an initial segment of $B$. If $A\sqsubseteq B$ and $A\neq B$, then we write $A\sqsubset B$ and say that $A$ is a proper initial segment of $B$.

Remark 5.13*.*

If a subset $A\subseteq\mathbb{T}_{k}$ does not contain sequences of [math]’s of unbounded length, then there is an $n\geq 0$ such that each node in $A$ has passing number $1$ at $c^{k}_{i}$, for some $i\in[-k+2,n]$. Such an $A$ cannot satisfy property $(A_{k})^{\tt tree}$ so it does not code $\mathcal{H}_{k}$; hence it is not strongly similar to $\mathbb{T}_{k}$. Thus, the leftmost path through any member of $\mathcal{T}_{k}$ is the infinite sequence of [math]’s. It follows that for $T\in\mathcal{T}_{k}$, the strong isomorphism $f_{T}:\mathbb{T}_{k}\rightarrow T$ must take each splitting node in $\mathbb{T}_{k}$ which is a sequence of [math]’s to a splitting node in $T$ which is a sequence of [math]’s. In particular, $f_{T}$ takes $c_{-k+2}^{k}$ to the ghost coding node $c_{-k+2}^{T}$ of $T$, which is a splitting node in $\mathbb{T}_{k}$ consisting of a sequence of [math]’s.

Next, we define what it means for a pre-clique to be witnessed inside a given subtree of $\mathbb{T}_{k}$. This sets the stage for the various Witnessing Properties to follow. The Strong Witnessing Property will be a structural characteristic of strong $\mathcal{H}_{k}$-coding trees. The Witnessing Property will be useful for extending a given finite subtree of some $T\in\mathcal{T}_{k}$ as should be possible according to the $K_{k}$-free Branching Criterion. Both will be utilized in following sections.

Definition 5.14** (Witnessing new pre-cliques in $T$).**

Let $T$ be a finite or infinite subtree of $\mathbb{T}_{k}$ with some coding and/or ghost coding nodes. Suppose that $X$ is a level subset of $T$ which has a new pre-$a$-clique at $l$, for some $a\in[3,k]$. We say that a set of coding and/or ghost coding nodes $\{c^{T}_{i}:i\in I\}$ in $T$, $|I|=a-2$, witnesses in $T$ that $X$ has a new pre-$a$-clique at $l$ if, letting $i_{*}=\max(I)$,

1. (1)

$\{c^{T}_{i}:i\in I\}$ codes an $(a-2)$-clique; 2. (2)

$|c^{T}_{i_{*}}|\geq l$ and $T$ has no critical nodes in the interval $[l,|c^{T}_{i_{*}}|)$; 3. (3)

For each $x\in X$, the node $y_{x}$ in $T\restriction|c^{T}_{i_{*}}|$ extending $x\restriction l$ has passing number $1$ at $c^{T}_{i}$, for each $i\in I$.

In this case, we write $WP_{a}(X;T)$. If $X\restriction l$ is a maximal new pre-$a$-clique and is witnessed by a set of coding nodes in $T$, we write $\mbox{WMP}_{a}(X;T)$.

Remark 5.15*.*

Note that the set $\{y_{x}:x\in X\}$ in (3) above is well-defined, since by (2), $T$ has no critical nodes (in particular no splitting nodes) in the interval $[l,|c^{T}_{i_{*}}|)$. The passing number of $y_{x}$ at $c_{i_{*}}^{T}$ is uniquely determined by $\mathbb{T}_{k}$ to be the passing number of $(y_{x})^{+}$ at $c_{i_{*}}^{T}$. We point out that there may be other sets of coding and/or ghost coding nodes in $\mathbb{T}_{k}$ witnessing $WP_{a}(X;T)$. However, any such set of witnesses in $T$ must contain $c^{T}_{i_{*}}$.

In the following, given a finite or infinite subtree $T$ of $\mathbb{T}_{k}$, recall that $\langle d^{T}_{m}:m\in M\rangle$ is the enumeration of the critical nodes in $T$ in order of increasing length. Here, either $M\in\mathbb{N}$ or $M=\mathbb{N}$. Enumerate the coding nodes in $T$ as $\langle c^{T}_{n}:n\in N\rangle$, where $N\in\mathbb{N}$ or $N=\mathbb{N}$. If $T$ has ghost coding nodes, enumerate these as $c^{T}_{n}$, where $n\in[-k+2,j]$ for some $j\leq-1$. The number $m_{n}$ is the index such that $d^{T}_{m_{n}}=c^{T}_{n}$.

Definition 5.16** (Strong Witnessing Property).**

A subtree $T$ of $\mathbb{T}_{k}$ has the Strong Witnessing Property if the following hold:

1. (1)

Each new pre-clique in $T$ takes place in some interval in $T$ of the form $(|d^{T}_{m_{n}-1}|,|c^{T}_{n}|]$. 2. (2)

Each new pre-clique in $T$ is witnessed in $T$.

Notice that the coding node $c^{T}_{n}$ in Definition 5.16 is obligated (by Definition 5.14) to be among the set of coding nodes witnessing $P_{a}(X;T)$. Further, in order to satisfy Definition 5.16, it suffices that the maximal new pre-$a$-cliques are witnessed in $T$, as this automatically guarantees that every new pre-$a$-clique is witnessed in $T$.

Definition 5.17** (Witnessing Property).**

A subtree $T$ of $\mathbb{T}_{k}$ has the Witnessing Property (WP) if each new pre-clique in $T$ of size at least two takes place in some interval in $T$ of the form $(|d^{T}_{m_{n}-1}|,|c^{T}_{n}|]$ and is witnessed by a set of coding nodes in $T$.

The idea behind the Witnessing Property is that we want a property strong enough to guarantee that the finite subtree $T$ of $\mathbb{T}_{k}$ can extend within $\mathbb{T}_{k}$ according to the $k$-FBC. However, each coding node is a pre-$k$-clique, and if $T$ is an antichain of coding nodes in $\mathbb{T}_{k}$ coding some finite $K_{k}$-free graph, we cannot require that all singleton pre-$k$-cliques be witnessed in $T$. The Witnessing Property will allow just the right amount of flexibility to achieve our Ramsey Theorem for antichains of coding nodes.

Lemma 5.18**.**

If $A$ is a subtree of $\mathbb{T}_{k}$ which has the (Strong) Witnessing Property and $A\cong B$, then $B$ has the (Strong) Witnessing Property.

Proof.

Given the hypotheses, let $f:B\rightarrow A$ be a strong isomorphism from $B$ to $A$. Suppose $X\subseteq B$ is a level set which has a new pre-$a$-clique, for some $a\in[3,k]$. Let $m$ be the index such that the new pre-$a$-clique in $X$ takes place in the interval $(|d^{B}_{m-1}|,|d^{B}_{m}|]$. Without loss of generality, assume that $X$ has a maximal new pre-$a$-clique in $B$ in this interval. In the case for WP, assume that $X$ has at least two members. Since $f$ is a strong isomorphism, $f[X]$ has a maximal new pre-$a$-clique in $A$ in the interval $(|d^{A}_{m-1}|,|d^{A}_{m}|]$. Since $A$ has the (Strong) Witnessing Property, $d^{A}_{m}$ must be a coding node in $A$; moreover, this coding node must be among the set of coding nodes in $A$ witnessing that $f[X]$ has a new pre-$a$-clique. Therefore, $d^{B}_{m}$ is a coding node, and $d^{B}_{m}$ is among the set of coding nodes witnessing that $X$ has a new pre-$a$-clique, since $f$ being a strong similarity map implies $f$ preserves coding nodes and passing numbers. Furthermore, each new pre-$a$-clique in $B$ (of size at least two in the case of WP) is witnessed in $B$. Hence, $B$ has the (Strong) Witnessing Property. ∎

Lemma 5.19**.**

Suppose $A,B$ are subtrees of $\mathbb{T}_{k}$ and that $A$ has the Strong Witnessing Property. Then $A\cong B$ if and only if $A\stackrel{{\scriptstyle s}}{{\sim}}B$ and $B$ also has the Strong Witnessing Property.

Proof.

For the forward direction, note that $A\cong B$ implies $A\stackrel{{\scriptstyle s}}{{\sim}}B$, by the definition of strongly isomorphic. If moreover, $A$ has the Strong Witnessing Property then Lemma 5.18 implies $B$ also has the Strong Witnessing Property.

Now suppose that $A\stackrel{{\scriptstyle s}}{{\sim}}B$ and both $A$ and $B$ have the Strong Witnessing Property. Let $f:A\rightarrow B$ be the strong similarity map. Suppose $X$ is a level set in $A$ which has a maximal new pre-$a$-clique, for some $a\in[3,k]$. Since $A$ has the Strong Witnessing Property, there is a set of coding nodes $C\subseteq A$ witnessing that $X$ has a new pre-$a$-clique. Furthermore, $l_{X}$ must be the length of some coding node in the set $C$. Since $f$ preserves coding nodes and passing numbers, it follows that $f[C]$ is a set of coding nodes in $B$ witnessing that $f[X]$ has a pre-$a$-clique. It remains to show that $f[X]$ has a new pre-$a$-clique and is maximal in $B$.

If $f[X]$ does not have a new pre-$a$-clique in $B$, then there is some critical node $d$ in $B$ below $f(c)$ such that $f[X]\restriction|d|$ has a new pre-$a$-clique in $B$, where $c$ is the longest coding node in $C$. Since $B$ satisfies the Strong Witnessing Property, this new pre-$a$-clique in $f[X]$ appears at some coding node in $B$ below $d$. Further, $f[X]$ must be witnessed by some set of coding nodes $D$ in $B$. But then $f^{-1}[D]$ is a set of coding nodes in $A$ witnessing a pre-$a$-clique in $X$. Since the longest length of a coding node in $f^{-1}[D]$ is shorter than $|c|$, the pre-$a$-clique in $X$ occurs first at some coding node below $c$, a contradiction to $X$ having a new pre-$a$-clique. Therefore, $f[X]$ is a new pre-$a$-clique in $B$.

If $f[X]$ is not maximal in $B$, then there is some level set $Z$ of nodes of length $l_{X}$ properly containing $f[X]$ which has a new pre-$a$-clique in $B$. Since $B$ has the Strong Witnessing Property, there is some set of coding nodes $D\subseteq B$ witnessing $Z$. Then $f^{-1}[D]$ witnesses that $f^{-1}[Z]$ is a pre-$a$-clique in $A$ properly containing $X$, contradicting the maximality of $X$ in $A$. Therefore, $f$ preserves maximal new pre-$a$-cliques, and hence is a strong isomorphism. Hence, $A\cong B$. ∎

The $K_{k}$-Free Branching Criterion from Definition 4.9 naturally gives rise to a version for skew trees. A tree $T\subseteq\mathbb{T}_{k}$ with (ghost) coding nodes $c_{n}$ for $n\in[-k+2,N)$, where $N\in\mathbb{N}$ or $N=\mathbb{N}$ and $c_{-k+2}$ is the stem of $T$, satisfies the $K_{k}$-Free Branching Criterion ($k$-FBC) if and only if the following holds: For $n\geq-k+2$, letting $u$ denote the node $c_{n+1}\restriction|c_{n}|$, each node $t\in T$ of length $|c_{n}|$ branches in $T$ before reaching length $|c_{n+1}|$ if and only if for each $I\subseteq[-k+2,n-1]$ of size $k-2$ for which the set $C=\{c_{i}:i\in I\}$ codes a $(k-2)$-clique and such that $u(|c_{i}|)=1$ for each $i\in I$, $t(|c_{i}|)=0$ for at least one $i\in I$.

Notice that if a skew tree $T$ satisfies the $k$-FBC, then Theorem 4.14 implies the following fact:

Observation 5.20**.**

Any skew tree satisfying the $K_{k}$-Free Branching Criterion in which the coding nodes are dense codes a copy of $\mathcal{H}_{k}$.

Lemma 5.21**.**

(1) If $T\subseteq\mathbb{T}_{k}$ is strongly similar to $\mathbb{T}_{k}$, then $T$ satisfies the $K_{k}$-Free Branching Criterion.

(2) If $T\subseteq\mathbb{T}_{k}$ is strongly similar to $\mathbb{T}_{k}$ and has the Strong Witnessing Property, then the strong similarity map from $\mathbb{T}_{k}$ to $T$ is a strong isomorphism, and hence $T$ is a member of $\mathcal{T}_{k}$.

Proof.

(1) follows in a straightforward manner from the definitions of $k$-FBC and strong similarity map, along with the structure of $\mathbb{T}_{k}$, as we now show. Suppose $T\subseteq\mathbb{T}_{k}$ is strongly similar to $T$, and let $f:\mathbb{T}_{k}\rightarrow T$ be the strong similarity map. Note that for each integer $n\geq-k+2$, $c^{T}_{n}=f(c^{k}_{n})$. Fix $n\geq-k+2$ and a node $t\in T\restriction l^{T}_{n}$ which does not extend to $c^{T}_{n+1}$. Then $s:=f^{-1}(t)$ is in $\mathbb{T}_{k}\restriction l^{k}_{n}$. Since $f$ is a strong similarity map, $s$ does not extend to the coding node $c^{k}_{n+1}$ in $\mathbb{T}_{k}$. Since $\mathbb{T}_{k}$ satisfies the $k$-FBC, $s$ splits in $\mathbb{T}_{k}$ before reaching the level of $c^{k}_{n+1}$ if and only if, letting $u=c^{k}_{n+1}\restriction(l_{n}^{k}+1)$, for each subset $I\subseteq[-k+2,n]$ of size $k-2$ such that $C=\{c^{k}_{i}:i\in I\}$ codes a $(k-2)$-clique and $u$ has passing number $1$ at each $c\in C$, there is some $c\in C$ at which $s^{+}$ has passing number [math]. Since $t=f(s)$ and $f$ is a strong similarity map, $t$ splits in $T$ before reaching the level of $c^{T}_{n+1}$ if and only if, letting $v=c^{T}_{n+1}\restriction(l^{T}_{n}+1)$, for each subset $I\subseteq[-k+2,n]$ of size $k-2$ for which $D=\{c^{T}_{i}:i\in I\}$ codes a $(k-2)$-clique and $v$ has passing number $1$ at each $c\in D$, there is some $c\in D$ at which $t^{+}$ has passing number [math].

For (2), if $T\subseteq\mathbb{T}_{k}$ is strongly similar to $\mathbb{T}_{k}$ and has the Strong Witnessing Property, then it follows from Lemma 5.19 that $T\cong\mathbb{T}_{k}$ since $\mathbb{T}_{k}$ has the Strong Witnessing Property. ∎

Lemma 5.22**.**

Every $T\in\mathcal{T}_{k}$ has the following properties:

1. (1)

$T\stackrel{{\scriptstyle s}}{{\sim}}\mathbb{T}_{k}$. 2. (2)

$T$* satisfies the $K_{k}$-Free Branching Criterion.* 3. (3)

$T$* has the Strong Witnessing Property.*

Proof.

(1) is immediate from the definition of $\mathcal{T}_{k}$. (2) follows from Lemma 5.21 part (1). (3) follows from (1) and Lemma 5.19. ∎

6. Extension Lemmas

Unlike Milliken’s strong trees, not every finite subtree of a strong $\mathcal{H}_{k}$-coding tree can be extended within that ambient tree to another member of $\mathcal{T}_{k}$, nor necessarily even to another finite tree of a desired configuration. This section provides structural properties of finite subtrees which are necessary and sufficient to extend to a larger tree of a particular strong similarity type. The first subsection lays the groundwork for these properties and the second subsection proves extension lemmas which are fundamental to developing Ramsey theory on strong $\mathcal{H}_{k}$-coding trees. The extension lemmas extend and streamline similar lemmas in [14], taking care of new issues that arise when $k\geq 4$. Furthermore, these lemmas lay new groundwork for general extension principles, with the benefit of a simpler proof of Theorem 7.2 than the proof of its instance for $\mathcal{H}_{3}$ in [14].

6.1. Free level sets

In this subsection, we provide criteria which will aid in the extension lemmas in Subsection 6.2. These requirements will guarantee that a finite subtree of a strong coding tree $T$ can be extended within $T$ to another strong coding tree.

Recall that given a tree $T\in\mathcal{T}_{k}$ and $m\in\mathbb{N}$, $T(m)$ denotes the set of nodes in $T$ of length equal to $|d^{T}_{m}|$, the length of the $m$-th critical node in $T$. The length of a (ghost) coding node $c^{T}_{n}$ is denoted $l^{T}_{n}$, which equals $|d^{T}_{m_{n}}|$. Thus, $T(m_{n})=T\restriction l^{T}_{n}$. Throughout this section, $n\geq-k+2$, and when we write “coding node” we are including ghost coding nodes.

Definition 6.1** (Free).**

Let $T\in\mathcal{T}_{k}$ be fixed. We say that a level set $X\subseteq\widehat{T}$ with length $l$ is free in $T$ if given $n$ least such that $l^{T}_{n}\geq l$, letting $Y$ consist of the leftmost extensions of members of $X$ in $T\restriction l^{T}_{n}$, then $Y$ has no new pre-$a$-cliques over $X$, for any $a\in[3,k]$.

In particular, any level set $X\subseteq T$ with length that of some coding node in $T$ is free in $T$.

Remark 6.2*.*

For $k=3$, this is equivalent to the concept of “$X$ has no pre-determined new parallel $1$’s in $T$” in [14].

Terminology 6.3**.**

For a level set $Y$ end-extending a level set $X$, we say that $Y$ has no new pre-cliques over $X$ if $Y$ has no new pre-$a$-cliques over $X$, for any $a\in[3,k]$.

Lemma 6.4**.**

Let $T\in\mathcal{T}_{k}$ be fixed, $X\subseteq\widehat{T}$ be a level set which is free in $T$. Then for any $l>l_{X}$, the set of leftmost extensions in $T$ of the nodes in $X$ to $T\restriction l$ contains no new pre-cliques over $X$. Furthermore, for any $n$ such that $l^{T}_{n}>l_{X}$, the leftmost extensions of $X$ in $T\restriction l^{T}_{n}$ have passing numbers [math] at $c^{T}_{n}$. It follows that any set of leftmost extensions of $X$ is free in $T$.

Proof.

This lemma follows from the fact that $T\cong\mathbb{T}_{k}$. To see this, let $f:\mathbb{T}_{k}\rightarrow T$ be the strong isomorphism witnessing that $T\in\mathcal{T}_{k}$, and let $j$ be least such that $l^{T}_{j}\geq l_{X}$. Let $n\geq j$ and $a\in[3,k]$ be given, and let $Y$ be the end-extension of $X$ in $T\restriction l^{T}_{n}$ consisting of the nodes which are leftmost extensions in $T$ of the nodes in $X$. Since $X$ is free in $T$, $Y$ has no new pre-$a$-cliques over $X$. Since $f^{-1}$ is a strong similarity map, $f^{-1}[Y]$ is the collection of leftmost extensions in $\mathbb{T}_{k}\restriction l^{k}_{n}$ of the level set $f^{-1}[X]$. In particular, $f^{-1}[Y]$ has no new pre-$a$-cliques in the interval $[l^{k}_{j},l^{k}_{n}]$. In particular, the passing numbers of members of $f^{-1}[Y]$ in this interval $[l^{k}_{j},l^{k}_{n}]$ are all [math]. Since $f$ is a strong isomorphism, $Y=f\circ f^{-1}[Y]$ has no new pre-$a$-cliques over $Y\restriction l_{X}$, and all passing numbers of the leftmost extensions of $X$ in $T$ are [math]. In particular, any set of leftmost extensions of $X$ in $T$ is free in $T$. ∎

An important property of $\mathcal{T}_{k}$ is that all of its members contain unbounded sequences of [math]’s.

Lemma 6.5**.**

Suppose $T\in\mathcal{T}_{k}$ and $s$ is a node in the leftmost branch of $T$. Then $s$ is a sequence of [math]’s.

Proof.

If not, then for some $n$, no node of $T$ extends $0^{(n)}$. Let $n$ be the least such integer. Then for each $t\in T\restriction n$, there is some coding node $c\in r_{n}(\mathbb{T}_{k})$ such that $t(|c|)=1$.

If there is a $t\in T\restriction n$ such that for all coding nodes $c^{T}_{i}$ in $T$ of length less than $n$, $t(|c^{T}_{i}|)=0$, then by the $K_{k}$-Free Branching Criterion of $T$, $t$ extends in $T$ to $k-1$ many coding nodes forming a $(k-1)$-clique. But this $(k-1)$-clique forms a $k$-clique with the vertex represented by any coding node in $r_{n}(\mathbb{T}_{k})$ for which $t(|c|)=1$, contradicting that $T\in\mathcal{T}_{k}$.

Otherwise, for each $t\in T\restriction n$, there is a coding node $c\in T$ with $|c|<n$ such that $t(|c|)=1$. But then $T$ cannot code a copy of $\mathcal{H}_{k}$, contradicting that $T\in\mathcal{T}_{k}$. ∎

Notation 6.6**.**

Let $A$ be a finite subtree of $\mathbb{T}_{k}$. We let $l_{A}$ denote the maximum of the lengths of the nodes in $A$, and let

[TABLE]

Thus, $\max(A)$ is a level set. If the maximal nodes in $A$ do not all have the same length, then $\max(A)$ will not contain all the maximal nodes in $A$.

Definition 6.7** (Finite strong coding tree).**

A finite subtree $A\subseteq\mathbb{T}_{k}$ is a finite strong coding tree if and only if $A\in\mathcal{AT}^{k}_{m+1}$ for $m$ such that either $d^{k}_{m}$ is a coding node or else $m=0$.

Lemma 6.8**.**

Given $T\in\mathcal{T}_{k}$, each finite strong coding tree $A$ contained in $T$ has the Strong Witnessing Property, and $\max(A)$ is free in $T$.

Proof.

Fix $T\in\mathcal{T}_{k}$ and let $A$ be a finite strong coding tree contained in $T$. If $A\in\mathcal{AT}_{0}^{k}$, then $A$ is the empty set, so the lemma vacuously holds. Otherwise, by Definition 6.7, $\max(A)$ contains a coding node, so $\max(A)$ is free in $T$. Further, $A\cong r_{m+1}(\mathbb{T}_{k})$ for some $m$ such that $d^{k}_{m}$ is a coding node. As $r_{m+1}(\mathbb{T}_{k})$ has the Strong Witnessing Property, Lemma 5.19 implies that $A$ also has the Strong Witnessing Property. ∎

6.2. Extension Lemmas

The next series of lemmas will be used extensively throughout the rest of the paper. As consequences, these lemmas ensure that every tree in $\mathcal{T}_{k}$ contains infinitely many subtrees which are also members of $\mathcal{T}_{k}$, and that for any $A\in\mathcal{AT}_{m}^{k}$ with $A\subseteq T\in\mathcal{T}_{k}$ and $\max(A)$ free in $T$, the set $r_{j}[A,T]$, defined in (40) of Definition 5.12, is infinite for each $j>m$.

Lemma 6.9**.**

Suppose $T\in\mathcal{T}_{k}$ and $X\subseteq T$ is a level set with $X$ free in $T$. Fix a subset $X^{\prime}\subseteq X$. Let $Y^{\prime}\subseteq T$ be any level set end-extending $X^{\prime}$ such that $Y^{\prime}$ is free in $T$. Let $Y^{\prime\prime}$ denote the set of leftmost extensions of $X\setminus X^{\prime}$ in $T\restriction l_{Y^{\prime}}$. Then $Y=Y^{\prime}\cup Y^{\prime\prime}$ is free in $T$, and any new pre-cliques in $Y$ occur in $Y^{\prime}$. In particular, if $Y^{\prime}$ has no new pre-cliques over $X^{\prime}$, then $Y$ has no new pre-cliques over $X$.

Proof.

Since $Y^{\prime}$ is free in $T$ and Lemma 6.4 implies that $Y^{\prime\prime}$ is free in $T$, it follows that $Y$ is free in $T$. Suppose that for some $a\in[3,k]$ there is a $Z\subseteq Y$ such that $Z$ has a new pre-$a$-clique in the interval $(l_{X},l_{Y}]$. Let $X^{\prime\prime}=X\setminus X^{\prime}$, $Z^{\prime}=Z\cap Y^{\prime}$, and $Z^{\prime\prime}=Z\cap Y^{\prime\prime}$. By Lemma 6.4, $Y^{\prime\prime}$ has no new pre-cliques over $X^{\prime\prime}$, so $Z^{\prime}$ must be non-empty.

Suppose toward a contradiction that also $Z^{\prime\prime}$ is non-empty. Let $l$ be the minimal length at which this new pre-$a$-clique occurs, and let $m$ be the least integer such that $|d^{T}_{m}|<l\leq|d^{T}_{m+1}|$. Since $T$ has the Strong Witnessing Property (recall Lemma 5.22), $d^{T}_{m+1}$ must be a coding node in $T$, say $c^{T}_{j}$. In the case that $l_{Y}<l^{T}_{j}$, by Lemma 6.4 we may extend the nodes in $Z$ leftmost to nodes in $T\restriction l^{T}_{j}$ without adding any new pre-cliques. Thus, without loss of generality, assume that $l_{Y}\geq l^{T}_{j}$. Since the new pre-$a$-clique $Z\restriction l$ must be witnessed in $T$ at the level of $c^{T}_{j}$, it follows that all nodes in $Z$ have passing number $1$ at $c^{T}_{j}$. Let $f$ be the strong isomorphism from $T$ to $\mathbb{T}_{k}$ and let $V$ denote $f[Z\restriction l^{T}_{j}]$. Then $V$ is a level set in $\mathbb{T}_{k}$ of length $l^{k}_{j}$. Since $f$ is a strong similarity map, it preserves passing numbers. Hence, all nodes in $V$ have passing number $1$ at $c^{k}_{j}$ in $\mathbb{T}_{k}$.

However, since $Y^{\prime\prime}$ consists of the leftmost extensions in $T$ of the nodes in $X^{\prime\prime}$, it follows that $f[Y^{\prime\prime}]$ is the set of leftmost extensions in $\mathbb{T}_{k}$ of $f[X^{\prime\prime}]$. Thus, each member of $f[Y^{\prime\prime}]$ has passing number [math] at $c_{j}^{k}$. Since $Z^{\prime\prime}\neq\emptyset$, also $f[Z^{\prime\prime}]\neq\emptyset$, so $V$ has at least one node with passing number [math] at $c^{k}_{j}$, contradicting the previous paragraph. Therefore, $Z^{\prime\prime}$ must be empty, so $Z$ resides entirely within $Y^{\prime}$. In particular, if $Y^{\prime}$ has no new pre-cliques over $X^{\prime}$, then $Y$ has no new pre-cliques over $X$. ∎

Lemma 6.10**.**

Suppose $s$ is a node in a strong coding tree $T\in\mathcal{T}_{k}$. If $n$ is least such that $|s|\leq l^{T}_{n}$, then there is splitting node $t$ in $T$ extending $s$ such that $|t|\leq l^{T}_{n+k}$. In particular, every strong coding tree is perfect.

Proof.

It suffices to work with $\mathbb{T}_{k}$, since each member of $\mathcal{T}_{k}$ is strongly isomorphic to $\mathbb{T}_{k}$. We make use here of the particular construction of $\mathbb{T}_{k}$ from Example 5.1.

Let $s$ be a node in $\mathbb{T}_{k}$, and let $n$ be least such that $l^{k}_{n}\geq|s|$. Let $p>n$ be least such that $p=i(k-1)$ for some $i\geq 1$, and let $s^{\prime}$ be the leftmost extension of $s$ in $\mathbb{T}_{k}\restriction(l^{k}_{p}+1)$. Note that $p<n+k$ and that $s^{\prime}$ has passing number [math] at $c^{k}_{p}$. By the construction of $\mathbb{T}_{k}$, $c^{k}_{p+1}$ in $\mathbb{T}_{k}$ will have passing number $1$ at precisely $c^{k}_{p-k+3},\dots,c^{k}_{p}$, and at no others. Let $v$ denote the truncation of $c^{k}_{p+1}$ to length $l^{k}_{p}+1$. The number of coding nodes in $\mathbb{T}_{k}$ at which both $s^{\prime}$ and $v$ have passing number $1$ is at most $k-3$. Therefore, $s^{\prime}$ and $v$ do not code a pre-$k$-clique. So by the $k$-FBC, $s^{\prime}$ extends to a splitting node $t$ in $\mathbb{T}_{k}$ before reaching the level of $c^{k}_{p+1}$. ∎

Given a set of nodes $Z\subseteq\mathbb{T}_{k}$, let $Z^{\wedge}$ denote the meet-closure of $Z$, which is the set of nodes $\{s\wedge t:s,t\in Z\}$. By the tree induced by $Z$ we mean the set of nodes $\{t\restriction|v|:t\in Z,\ v\in Z^{\wedge}\}$.

Lemma 6.11**.**

Suppose $A$ is a finite subtree of some strong coding tree $T\in\mathcal{T}_{k}$ with $\max(A)$ free in $T$. Let $X$ be any nonempty subset of $\max(A)$, and let $Z$ be any subset of $\max(A)\setminus X$. Let $\{s_{i}:i<\tilde{i}\}$ be any enumeration of $X$ and suppose $l\geq l_{A}$ is given. Then there exist $l_{*}>l$ and extensions $t_{i}^{0},t_{i}^{1}\supset s_{i}$ ($i<\tilde{i}$) and $t_{z}\supset z$ ($z\in Z$), each in $T\restriction l_{*}$, such that letting

[TABLE]

and letting $B$ denote the tree induced by $A\cup Y$, the following hold:

1. (1)

The splitting in $B$ above level $l_{A}$ occurs in the order of the enumeration of $X$. Thus, for $i<i^{\prime}<\tilde{i}$, $|t_{i}^{0}\wedge t_{i}^{1}|<|t_{i^{\prime}}^{0}\wedge t_{i^{\prime}}^{1}|$. 2. (2)

$Y$* has no new pre-cliques over $\max(A)$ and is free in $T$.*

Proof.

If $l_{A}$ is not the level of some coding node in $T$, begin by extending each member of $X$ leftmost in $T$ to the level of the very next coding node in $T$. In this case, abuse notation and let $X$ denote this set of extensions. Since $\max(A)$ is free in $T$, this adds no new pre-cliques over $\max(A)$.

By Lemma 6.10, every node in $X$ extends to a splitting node in $T$. Let $s_{0}^{*}$ be the splitting node of least length in $T$ extending $s_{0}$, and let $c^{T}_{n_{0}}$ be the coding node in $T$ of least length above $|s_{0}^{*}|$. Extend all nodes in $\{s_{i}:1\leq i<\tilde{i}\}$ leftmost in $T$ to length $l^{T}_{n_{0}}$, and label their extensions $\{s^{1}_{i}:1\leq i<\tilde{i}\}$. Given $1\leq p<\tilde{i}$ and the nodes $\{s^{p}_{i}:p\leq i<\tilde{i}\}$, let $s^{*}_{p}$ be the splitting node of least length in $T$ extending $s_{p}^{p}$, and let $c^{T}_{n_{p}}$ be the coding node in $T$ of least length above $|s_{p}^{*}|$. If $p<\tilde{i}-1$, then extend all nodes in $\{s^{p}_{i}:p+1\leq i<\tilde{i}\}$ leftmost in $T$ to length $l^{T}_{n_{p}}$, and label these $\{s^{p+1}_{i}:p+1\leq i<\tilde{i}\}$.

When $p=\tilde{i}-1$, let $n=n_{\tilde{i}-1}$ and for each $i<\tilde{i}$ and $j<2$, let $t_{i}^{j}$ be the leftmost extension in $T$ of ${s^{*}_{i}}^{\frown}j$ to length $l^{T}_{n}$. For each $z\in Z$, let $t_{z}$ be the leftmost extension in $T$ of $z$ to length $l^{T}_{n}$. This collection of nodes composes the desired set $Y$. By Lemma 6.9, $Y$ has no new pre-cliques over $\max(A)$. $Y$ is free in $T$ since the nodes in $T$ have the length of a coding node in $T$. ∎

Convention 6.12**.**

Recall that when working within a strong coding tree $T\in\mathcal{T}_{k}$, the passing numbers at coding nodes in $T$ are completely determined by $T$; in fact, they are determined by $\mathbb{T}_{k}$. For a finite subset $A$ of $T$ such that $l_{A}$ equals $l_{n}^{T}$ for some $n<\omega$, we shall say that $A$ has the (Strong) Witnessing Property if and only if the extension $A\cup\{s^{+}:s\in\max(A)\}$ has the (Strong) Witnessing Property.

The notion of a valid subtree is central to the constructions in this paper.

Definition 6.13** (Valid).**

Suppose $T\in\mathcal{T}_{k}$ and let $A$ be a finite subtree of $T$. We say that $A$ is valid in $T$ if and only if $A$ has the Witnessing Property and $\max(A)$ is free in $T$.

The next Lemma 6.14 shows that given a valid subtree of a strong coding tree $T$, any of its maximal nodes can be extended to some coding node $c_{n}^{T}$ in $T$ while the rest of the maximal nodes can be extended to length $l_{n}^{T}$ so that their passing numbers are anything desired, subject only to the $K_{k}$-Free Criterion.

Lemma 6.14** (Passing Number Choice).**

Fix $T\in\mathcal{T}_{k}$ and a finite subtree $A$ with $\max(A)$ free in $T$. Let $\{s_{i}:i<\tilde{i}\}$ be any enumeration of $\max(A)$, and fix some $d<\tilde{i}$. To each $i\in\tilde{i}\setminus\{d\}$ associate an $\varepsilon_{i}\in\{0,1\}$, with the stipulation that $\varepsilon_{i}$ must equal [math] if $\{s_{i},s_{d}\}$ has a pre-$k$-clique. In particular, $\varepsilon_{d}=0$.

Then given any $j$, there is an $n\geq j$ such that the coding node $c^{T}_{n}$ extends $s_{d}$, and there are extensions $u_{i}\supseteq s_{i}$, $i\in\tilde{i}\setminus\{d\}$, in $T\restriction l_{n}^{T}$ such that, letting $u_{d}$ denote $c_{n}^{T}$ and letting $Y=\{u_{i}:i<\tilde{i}\}$, the following hold:

1. (1)

Each $u_{i}$ has passing number $\varepsilon_{i}$ at $u_{d}$. 2. (2)

Any new pre-cliques among subsets of $Y$ (except possibly for the singleton $\{s_{d}\}$) have their first instances occurring in the interval $(|d^{T}_{m_{n}-1}|,l^{T}_{n}]$. 3. (3)

If $A$ has the Witnessing Property, then so does $A\cup Y$. Thus, if $A$ is valid, then $A\cup Y$ is also valid. 4. (4)

If $A$ has the Strong Witnessing Property and $s_{d}$ has a pre-$(k-1)$-clique witnessed by coding nodes in $A$, then $A\cup Y$ has the Strong Witnessing Property.

Proof.

Assume the hypotheses in the first paragraph of the lemma. Let $m$ be least such that $l^{T}_{m}\geq l_{A}$, and for each $i<\tilde{i}$, let $s^{\prime}_{i}$ be the leftmost extension of $s_{i}$ in $T$ of length $l_{m}^{T}$. Since $\max(A)$ is free in $T$, the set $\{s^{\prime}_{i}:i<\tilde{i}\}$ has no new pre-cliques over $A$. Given $j$, take $n$ minimal above $\max(j,m+1)$ such that $c_{n}^{T}\supseteq s^{\prime}_{d}$, and let $u_{d}=c_{n}^{T}$. Such an $n$ exists, as the coding nodes in any strong coding tree are dense in that tree, by its strong similarity to $\mathbb{T}_{k}$. For $i\neq d$, extend $s^{\prime}_{i}$ via its leftmost extension in $T$ to length $l_{n-1}^{T}$ and label it $t_{i}$. By Lemma 6.9, $\{t_{i}:i\in\tilde{i}\setminus\{d\}\}\cup\{u_{d}\restriction l^{T}_{n-1}\}$ has no new pre-cliques over $\{s^{\prime}_{i}:i<\tilde{i}\}$, with the possible exception of the singleton $u_{d}\restriction l^{T}_{n-1}$, so (2) of the Lemma holds.

For $i\in\tilde{i}\setminus\{d\}$ with $\varepsilon_{i}=0$, let $u_{i}$ be the leftmost extension of $t_{i}$ of length $l_{n}^{T}$. For $i<\tilde{i}$ with $\varepsilon_{i}=1$, by our assumption, $\{t_{i},u_{d}\restriction l^{T}_{n-1}\}$ has no pre-$k$-cliques, and their extensions to length $l^{T}_{n-1}$ have no new pre-cliques by Lemma 6.9. By the $k$-Free Branching Criterion of $T$, $t_{i}$ splits in $T$ before reaching the level of $c^{T}_{n}$. Let $u_{i}$ be the rightmost extension of $t_{i}$ to length $l_{n}^{T}$. Then for each $i<\tilde{i}$, the passing number of $u_{i}$ at $u_{d}$ is $\varepsilon_{i}$. Thus, (1) holds.

Suppose now that $A$ has the Witnessing Property. Let $Y=\{u_{i}:i<\tilde{i}\}$. Since the nodes in $Y$ have the length of the coding node $u_{d}$, $Y$ is free in $T$. By construction, any new pre-cliques in $Y$ over $A$ occur in the interval $(l^{T}_{n-1},l^{T}_{n}]$. Since $T$ has the Strong Witnessing Property, any new pre-cliques of $Y$ in the interval $(l^{T}_{n-1},l^{T}_{n}]$ must actually occur in the interval $(|d^{T}_{m_{n-1}}|,l^{T}_{n}]$. It remains to show that any new pre-cliques of size at least two in $Y$ over $A$ in the interval $(|d^{T}_{m_{n-1}}|,l^{T}_{n}]$ are witnessed by coding nodes in $A\cup Y$. We now show slightly more.

Suppose $I\subseteq\tilde{i}$, where $d\not\in I$, and $\{u_{i}:i\in I\}$ has a new pre-$a$-clique over $A$, for some $a\in[3,k]$. Let $Z$ denote $\{u_{i}:i\in I\}$ and let $l$ be least such that $Z\restriction l$ is a pre-$a$-clique, and note that $l$ must be in the interval $(|d^{T}_{m_{n}-1}|,l_{n}^{T}]$. Since $T$ has the Strong Witnessing Property, there is some set of coding nodes $C\subseteq T$ witnessing $Z\restriction l$. As $u_{d}$ is the least coding node in $T$ above $Z\restriction l$, $u_{d}$ must be in $C$, again by the Strong Witnessing Property of $T$. It follows that each node in $Z$ must have passing number $1$ at $u_{d}$.

If $a=3$, then the coding node $u_{d}$, which is contained in $Y$, witnesses the pre-$3$-clique in $Z$. Now suppose that $a\geq 4$. Then $C\setminus\{u_{d}\}$ witnesses that $Z^{\prime}=Z\cup\{u_{d}\}$ has a pre-$(a-1)$-clique. The $l^{\prime}$ at which $Z^{\prime}\restriction l^{\prime}$ is a new pre-$(a-1)$-clique must be below $|d^{T}_{m_{n}-1}|$, since $T$ cannot witness it at the level of $u_{d}$. Since $Y\setminus\{u_{d}\}$ has no new pre-cliques over $A$ in the interval $(l_{A},|d^{T}_{m_{n}-1}|]$, it must be that $l^{\prime}\leq l_{A}$. Since $Z^{\prime}\restriction l_{A}$ has size at least two and is contained in $A$, the Witnessing Property of $A$ implies that there is a set of coding nodes $C^{\prime}$ contained in $A$ witnessing the pre-$(a-1)$-clique $Z\restriction l^{\prime}$. Then $C^{\prime}\cup\{u_{d}\}$ is contained in $A\cup Y$ and witnesses the pre-$a$-clique $Z$.

Now, suppose that $d\in I\subseteq\tilde{i}$, $I$ has size at least two, and $\{u_{i}:i\in I\}$ has a new pre-clique over $A$. We will show that this is impossible. We point out that in the interval $(l^{T}_{n-1}+1,l_{n}^{T})$, the coding node $u_{d}$ has no new pre-cliques with any other node in $T$ of length $l^{T}_{n}$, (for $T$ has the Strong Witnessing Property and such a new pre-clique could not be witnessed in $T$). Since for each $i\neq d$, the node $u_{i}\restriction l^{T}_{n-1}$ is the leftmost extension of $s_{i}$ in $T$, it follows that for each $l\in(l_{A},l^{T}_{n-1}]$, the set $\{u_{i}\restriction l:i\in I\}$ has no new pre-cliques of size two or more. Thus, any new pre-clique occuring among $\{u_{i}:i\in I\}$ above $A$ must exclude $u_{d}$. It follows that $A\cup Y$ has the Witnessing Property, so (3) holds.

We have already shown that, assuming that $A$ has the Witnessing Property, $u_{d}$ is not in any subset of $Y$ of size at least two which has a new pre-clique over $A$, and that for $i\neq d$, any new pre-clique in $\{u_{i}\}$ over $A$ is witnessed in $A\cup Y$. Assuming the premise of (4), $s_{d}$ has a pre-$(k-1)$-clique witnessed by coding nodes in $A$. Thus, the Strong Witnessing Property of $A$ carries over to $A\cup Y$. ∎

The next lemma provides conditions under which a subtree of a strong coding tree can be extended to another subtree with a prescribed strong similarity type. This will be central to the constructions involved in proving the Ramsey theorems for strong coding trees as well as in further sections.

Lemma 6.15**.**

Suppose $A$ is a finite subtree of a strong coding tree $T\in\mathcal{T}_{k}$ with $\max(A)$ free in $T$. Fix any member $u\in\max(A)$. Let $X$ be any subset of $\max(A)$ such that for each $s\in X$, the pair $\{s,u\}$ has no pre-$k$-cliques, and let $Z$ denote $\max(A)\setminus(X\cup\{u\})$. Let $l\geq l_{A}$ be given.

Then there is an $l_{*}>l$ and there are extensions $u_{*}\supset u$, $s_{*}^{0},s_{*}^{1}\supset s$ for all $s\in X$, and $s_{*}\supset s$ for all $s\in Z$, each of length $l_{*}$, such that, letting

[TABLE]

and letting $B$ be the tree induced by $A\cup Y$, the following hold:

1. (1)

$u_{*}$* is a coding node.* 2. (2)

For each $s\in X$ and $i\leq 1$, the passing number of $s_{*}^{i}$ at $u_{*}$ is $i$. 3. (3)

For each $s\in Z$, the passing number of $s_{*}$ at $u_{*}$ is [math]. 4. (4)

Splitting among the extensions of the $s\in X$ occurs in reverse lexicographic order: For $s$ and $t$ in $X$, $|s_{*}^{0}\wedge s_{*}^{1}|<|t^{0}_{*}\wedge t^{1}_{*}|$ if and only if $s_{*}>_{\mathrm{lex}}t_{*}$. 5. (5)

There are no new pre-cliques among the nodes in $X$ below the length of the longest splitting node in $B$ below $u_{*}$.

If moreover $A$ has the (Strong) Witnessing Property, then $B$ also has the (Strong) Witnessing Property.

Proof.

Since $\max(A)$ is free in $T$, apply Lemma 6.11 to extend $\max(A)$ to have splitting nodes in the desired order without adding any new pre-cliques and so that this extension is free in $T$. Then apply Lemma 6.14 to extend to a level with a coding node and passing numbers as prescribed, with the extension being valid in $T$. Lemma 6.14 also guarantees that we can construct such a $B$ with the (Strong) Witnessing Property, provided that $A$ has the (Strong) Witnessing Property. ∎

This immediately yields the main extension theorem of this section.

Theorem 6.16**.**

Suppose $T\in\mathcal{T}_{k}$, $m<\omega$, and $A$ is a member of $\mathcal{AT}_{m}$ with $\max(A)$ free in $T$. Then the set $r_{m+1}[A,T]$ is infinite. In particular, for each $l<\omega$, there is a member $B\in r_{m+1}[A,T]$ with $\max(B)$ free in $T$ and $l_{B}\geq l$. Furthermore, $[A,T]$ is infinite, and for each strictly increasing sequence of integers $(l_{j})_{j>m}$, there is a member $S\in[A,T]$ such that $|d^{S}_{j}|>l_{j}$ and $\max(r_{j}(S))$ is free in $T$, for each $j>m$.

Proof.

Recall that every member of $\mathcal{AT}_{m}$ has the Strong Witnessing Property. The first part of the theorem follows from Lemmas 6.11 and 6.14. The second part follows from Lemma 6.15. ∎

The final lemmas of this section set up for constructions in the main theorem of Section 7.

Lemma 6.17**.**

Suppose $T\in\mathcal{T}_{k}$, $X\subseteq T$ is a level set containing a coding node $c^{T}_{j}$, and $X^{\prime}\cup X^{\prime\prime}$ is a partition of $X$ with $c^{T}_{j}\in X^{\prime}$. Suppose further that $j<n$, $c^{T}_{n}$ extends $c^{T}_{j}$, and $c_{n}^{T}\in Y^{\prime}\subseteq T\restriction l^{T}_{n}$ end-extends $X^{\prime}$ with the following properties: $Y^{\prime}$ has no new pre-cliques over $X^{\prime}$, and each node in $Y^{\prime}$ has the same passing number at $c^{T}_{n}$ as it does at $c^{T}_{j}$. Then there is a level set $Y^{\prime\prime}\subseteq T\restriction l^{T}_{n}$ end-extending $X^{\prime\prime}$ such that each node in $Y^{\prime\prime}$ has the same passing number at $c^{T}_{n}$ as it does at $c^{T}_{j}$, and $Y=Y^{\prime}\cup Y^{\prime\prime}$ has no new pre-cliques over $X$.

Proof.

If $n>j+1$, first extend the nodes in $X^{\prime\prime}$ leftmost in $T$ to length $l^{T}_{n-1}$, and label this set of nodes $Y^{\prime\prime}\restriction l^{T}_{n-1}$. By Lemma 6.9 $Y^{\prime\prime}\restriction l^{T}_{n-1}\cup Y^{\prime}\restriction l^{T}_{n-1}$ has no new pre-cliques over $X$. Apply Lemma 6.14 to extend the nodes in $Y^{\prime\prime}\restriction l^{T}_{n-1}$ to $Y^{\prime\prime}\in T\restriction l^{T}_{n}$ such that for each node $t\in Y^{\prime\prime}$, $t$ has the same passing number at $c^{T}_{n}$ as it does at $c^{T}_{j}$. Let $Y=Y^{\prime}\cup Y^{\prime\prime}$.

Suppose towards a contradiction that for some $a\in[3,k]$, there is a new pre-$a$-clique $Z\subseteq Y$ above $X$. If $a=3$, then $c^{T}_{n}$ witnesses this pre-$3$-clique. Since each node in $Y$ has the same passing number at $c^{T}_{n}$ as it does at $c^{T}_{j}$, it follows that $Z\restriction l^{T}_{j}$ has a pre-$3$-clique which is witnessed by $c^{T}_{j}$. Thus, $Z$ was not new over $X$.

Now suppose that $a\geq 4$. Then $Z\cup\{c^{T}_{n}\}$ has a pre-$b$-clique, where $b=a-1$. Since $Z$ is a new pre-$a$-clique and $T\cong\mathbb{T}_{k}$, it must be that the the level where the pre-$b$-clique in $Z\cup\{c^{T}_{n}\}$ is new must be at some $l\leq l^{T}_{n-1}$. Since $Y$ has no new pre-cliques in the interval $(l^{T}_{j},l^{T}_{n-1}]$, this $l$ must be less than or equal to $l^{T}_{j}$. Since the passing numbers of members in $Y$ are the same at $c^{T}_{n}$ as they are at $c^{T}_{j}$, it follows that $Z\cup\{c^{T}_{j}\}$ has a pre-$b$-clique. This pre-$b$-clique must occur at some level strictly below $l^{T}_{j}$, since the passing number of the coding node $c^{T}_{j}$ at itself is [math]. Hence, $Z\restriction l^{T}_{j}\cup\{c^{T}_{j}\}$ is a pre-$a$-clique. Therefore, $Z$ is not a new pre-$a$-clique over $X$, a contradiction. ∎

Lemma 6.18**.**

Suppose $T\in\mathcal{T}_{k}$, $m\in\mathbb{N}$, and $B\in r_{m+1}[0,T]$ with $\max(B)$ free in $T$. Let $x_{*}$ be the critical node of $\max(B)$, let $X\subseteq\max(B)$ with $x_{*}\in X$, and let $X^{\prime}=\max(B)\setminus X$. Suppose that $Y$ end-extends $X$ into $T$ so that $Y$ has no new pre-cliques over $X$, $Y$ is free in $T$, and the critical node $x_{*}$ is extended to the same type of critical node $y_{*}$ in $Y$. If $x_{*}$ is a coding node, assume that for each $y\in Y$, the passing number of $y$ at $y_{*}$ is the same as the passing number of $y$ at $x_{*}$. Then there is a level set $Y^{\prime}$ end-extending $X^{\prime}$ in $T$ to length $l_{Y}$ such that $r_{m}(B)\cup(Y\cup Y^{\prime})$ is a member of $r_{m+1}[r_{m}(B),T]$.

Proof.

Suppose first that $x_{*}$ is a splitting node. By Lemma 6.9, letting $Y^{\prime}$ be the level set of leftmost nodes in $T\restriction|y_{*}|$, we see that $Y\cup Y^{\prime}$ is free in $T$ and has no new pre-cliques over $\max(B)$. In particular, $r_{m}(B)\cup(Y\cup Y^{\prime})$ is a member of $r_{m+1}[r_{m}(B),T]$.

Now suppose that $x_{*}$ is a coding node. Let $n$ be the integer such that $y_{*}=c^{T}_{n}$. Then $l_{X}\leq l^{T}_{n-1}$. Let $W^{\prime}$ denote the leftmost extensions of the nodes in $X^{\prime}$ in $T\restriction l^{T}_{n-1}$. Again by Lemma 6.9, the set $W^{\prime}\cup(Y\restriction l^{T}_{n-1})$ has no new pre-cliques over $B$. For $i\in 2$, let $W^{\prime}_{i}$ be the set of those $w\in W^{\prime}$ which have passing number $i$ at $x_{*}$. Note that for each $w\in W^{\prime}_{1}$, the set $\{w\restriction l_{X},x_{*}\}$ has no pre-$k$-clique, and since no new pre-cliques are added between the levels of $l_{X}$ and $l^{T}_{n-1}$, the set $\{w,y_{*}\restriction l^{T}_{n-1}\}$ has no pre-$k$-clique. Since $T$ satisfies the $K_{k}$-Free Branching Criterion, each $w\in W^{\prime}_{1}$ can be extended to a node $y\in T\restriction l_{Y}$ with passing number $1$ at $y_{*}$. Extend each node in $W^{\prime}_{0}$ leftmost in $T$ to length $l_{Y}$. Let $Y^{\prime}=W^{\prime}_{0}\cup W^{\prime}_{1}$. Then $Y^{\prime}$ end-extends $W^{\prime}$ which end-extends $X^{\prime}$, and each $y\in Y^{\prime}$ has the same passing number at $y_{*}$ as it does at $x_{*}$.

We claim that $Y\cup Y^{\prime}$ has no new pre-cliques over $B$. Suppose towards a contradiction that $Z\subseteq Y\cup Y^{\prime}$ is a new pre-$a$-clique above $B$, for some $a\in[3,k]$. Since $Z\restriction l^{T}_{n-1}$ has no new pre-cliques over $B$, this new pre-$a$-clique must take place at some level $l\in(l^{T}_{n-1},l_{Y}]$. Since $T$ has the Strong Witnessing Property, $l$ must be in the interval $(|d|,l_{Y}]$, where $d$ is the longest splitting node in $T$ of length less than $l_{Y}$, and $y_{*}$ must be among the set of witnessing coding nodes in $T$ for this new pre-$a$-clique. If $a=3$, then $y_{*}$ witnesses the pre-$3$-clique $Z$. But then $Z\restriction l_{X}$ must also be a pre-$3$-clique, since the passing numbers at $y_{*}$ are the same as at $x_{*}$, and $x_{*}$ witnesses the pre-$3$-clique $Z\restriction l_{X}$. Hence, $Z$ is not new over $B$. Now suppose that $a\geq 4$. Then $Z\cup\{y_{*}\}$ has a new pre-$(a-1)$-clique at some level $l^{\prime}<l$. Since $T$ has the Strong Witnessing Property this new pre-$(a-1)$-clique must be witnessed in $T$. Since $T$ has no new pre-cliques in $(l^{T}_{n-1},|d|]$ and $Z\restriction l^{T}_{n-1}$ has no new pre-cliques over $B$, it must be that $l^{\prime}\leq l_{X}$. Therefore, the minimal level of a pre-$(a-1)$-clique in $Z\cup\{y_{*}\}$ is at some level in $B$. Since $B$ has the Strong Witnessing Property, this is witnessed in $B$. Since $y_{*}\supseteq x_{*}$ and each $z\in Z$ has the same passing number at $y_{*}$ as at $x_{*}$, $x_{*}$ cannot be a witness of the pre-$(a-1)$-clique in $Z\cup\{y_{*}\}$. Therefore, $Z\cup\{y_{*}\}$ must be witnessed in $r_{m}(B)$, say by coding nodes $\{c^{B}_{i_{j}}:j<a-3\}$, where $i_{a-4}<l^{B}_{m}$. But then $\{x_{*}\}\cup\{c^{B}_{i_{j}}:j<a-3\}$ witnesses the pre-$a$-clique $Z$. Hence, $Z$ is not new over $B$.

Now we will show that $Y\cup Y^{\prime}$ has no new pre-cliques over $r_{m}(B)$. Suppose $Z\subseteq Y\cup Y^{\prime}$ has a pre-$a$-clique, for some $a\in[3,k]$. Since this pre-$a$-clique is not new over $B$, there is some $l\leq l_{X}$ where $Z\restriction l$ is a new pre-$a$-clique in $B$. Since $B$ has the Strong Witnessing Property, there are some coding nodes $c_{i_{0}}^{B},\dots c_{i_{a-3}}^{B}$ in $B$ witnessing $Z\restriction l$. If $i_{a-3}<m$, then these witnesses are in $r_{m}(B)$. Now suppose that $i_{a-3}=m$. Note that $y_{*}\supseteq x_{*}=c^{B}_{m}$. Thus, $\{y_{*}\}\cup\{c^{B}_{i_{j}}:j<a-3\}$ forms a pre-$(a-1)$-clique which witnesses $Z$. Therefore, $r_{m}(B)\cup Y^{\prime}\cup Y$ has the Strong Witnessing Property. Since it is strongly similar to $B$, $r_{m}(B)\cup Y^{\prime}\cup Y$ is a member of $r_{m+1}[r_{m}(B),T]$ by Lemma 5.19. ∎

Remark 6.19*.*

As was remarked for $\mathcal{T}_{3}$ in [14], each space $(\mathcal{T}_{k},\leq,r)$, $k\geq 3$, satisfies Axioms A.1, A.2, and **A.3(1) **of Todorcevic’s axioms in Chapter 5 of [54] guaranteeing a topological Ramsey space, and it is routine to check this. However, Axiom **A.3(2) **does not hold. The pigeonhole principle, Axiom A.4, holds exactly when the finite subtree is valid inside the given strong coding tree; this will follow from Theorems in Section 7 and 8.

7. Halpern-Läuchli-style Theorems for strong coding trees

The Ramsey theory content for strong coding trees begins in this section. The ultimate goal is to obtain Theorem 9.9. This is a Ramsey theorem for colorings of strictly similar (Definition 9.4) copies of any given finite antichain of coding nodes, as these are the structures which will code finite triangle-free graphs.

Phase II of the paper takes place in this and the next section. Theorem 7.2 is a Halpern-Läuchli-style theorem for colorings of level sets extending a finite valid subtree of some strong coding tree. Its proof begins with Harrington’s forcing proof of Theorem 3.8 as a rough template, but involves new forcings and new arguments necessitated by the $k$-clique-free nature of Henson graphs. This is a major step toward proving a Milliken-style theorem for strong coding trees, but it is not enough: in the case when there is a coding node in the level sets being colored, Theorem 7.2 proves homogeneity on level sets extending some fixed set of nodes, but does not obtain homogeneity overall. We will have a third Halpern-Läuchli-style theorem in Section 8, which involves ideas unprecedented to our knowledge. This Lemma 8.8 will use a third type of forcing. Then using much induction on Theorem 7.2 and Lemma 8.8, we will prove the Main Ramsey Theorem for Strictly Witnessed (see Definition 8.1) finite subtrees of a given strong coding tree. This Theorem 8.3 is the main theorem of Phase II of the paper.

Theorem 7.2 encompasses colorings of two different types of level set extensions of a fixed finite tree: The level set either contains a splitting node (Case (a)) or a coding node (Case (b)). In Case (a), we obtain a direct analogue of the Halpern-Läuchli Theorem. In Case (b), we obtain a weaker version of the Halpern-Läuchli Theorem, which is later strengthened to the direct analogue in Lemma 8.8. The proof given here essentially follows the outline of the proof of Theorem 5.2 in [14], but our argument is now more streamlined, due to having proved more general extension lemmas in Section 5.

Let $k\geq 3$ be fixed, and fix the following terminology and notation. Given subtrees $U,V$ of $\mathbb{T}_{k}$ with $U$ finite, we write $U\sqsubseteq V$ if and only if $U=\{v\in V:|v|\leq l_{U}\}$; in this case we say that $V$ extends $U$, or that $U$ is an initial subtree of $V$. We write $U\sqsubset V$ if $U$ is a proper initial subtree of $V$. Recall the following notation from Definition 5.12 of the space $(\mathcal{T}_{k},\leq,r)$: $S\leq T$ means that $S$ and $T$ are members of $\mathcal{T}_{k}$ and $S$ is a subtree of $T$. Given $A\in\mathcal{AR}_{m}$ for some $m$, $[A,T]$ denotes the set of all $S\leq T$ such that $S$ extends $A$. We now begin setting up for the two possible cases before stating the theorem.

The Set-up for Theorem 7.2.** Let $T\in\mathcal{T}_{k}$ be given, and let $A$ be a finite valid subtree of $T$. $A$ is allowed to have terminal nodes at different levels. In order to simplify notation in the proof, without loss of generality, we assume that $0^{(l_{A})}$ is in $A$. Let $A^{+}$ denote the set of immediate extensions in $\widehat{T}$ of the members of $\max(A)$; thus,**

[TABLE]

Note that $A^{+}$ is a level set of nodes of length $l_{A}+1$. Let $A_{e}$ be a subset of $A^{+}$ containing $0^{(l_{A}+1)}$ and of size at least two. (If $A^{+}$ has only one member, then $\max(A)$ consists of one non-splitting node of the form $0^{(l)}$ for some $l$, and the theorem in this section does not apply.) Suppose that $\tilde{X}$ is a level set of nodes in $T$ extending $A_{e}$ so that $A\cup\tilde{X}$ is a finite valid subtree of $T$. Assume moreover that $0^{(l_{\tilde{X}})}$ is a member of $\tilde{X}$, so that the node $0^{(l_{A})}$ in $A_{e}$ is extended by $0^{(l_{\tilde{X}})}$ in $\tilde{X}$. There are two possibilities:

1. 1. Case (a).

$\tilde{X}$** contains a splitting node.**

1. 1. Case (b).

$\tilde{X}$** contains a coding node.**

In both cases, define

[TABLE]

The next lemma shows that seemingly weaker properties suffice to guarantee that a level set is in $\operatorname{Ext}_{T}(A,\tilde{X})$.

Lemma 7.1**.**

Let $X$ be a level set in $T$ extending $\tilde{X}$. Then $X\in\operatorname{Ext}_{T}(A,\tilde{X})$ if and only if $X$ is free in $T$, $A\cup X$ is strongly similar to $A\cup\tilde{X}$, and $X$ has no new pre-cliques over $A\cup\tilde{X}$. Moreover, $X\in\operatorname{Ext}_{T}(A,\tilde{X})$ implies that $A\cup X$ has the Witnessing Property.

Proof.

The forward direction follows from the definition of $\operatorname{Ext}_{T}(A,\tilde{X})$. If $X\in\operatorname{Ext}_{T}(A,\tilde{X})$, then there is a strong isomorphism, say $f:A\cup\tilde{X}\rightarrow A\cup X$. Then $f$ is a strong similarity map and moreover, $f$ takes $\tilde{X}$ to $X$. Since $X$ extends $\tilde{X}$ and $f$ maps the new pre-cliques of $\tilde{X}$ over $A$ to the new pre-cliques of $X$ over $A$, $X$ must have no new pre-cliques over $A\cup\tilde{X}$. Note that $X$ is free in $T$, since $A\cup X$ is valid in $T$.

Now suppose that $X\sqsupseteq\tilde{X}$ is as in the second part of the statement. Since $X$ is free in $T$, $A\cup X$ is valid in $T$. $A\cup X$ being strongly similar to $A\cup\tilde{X}$ implies that the strong similarity map $g:A\cup\tilde{X}\rightarrow A\cup X$ takes $\tilde{X}$ to $X$. Since $X$ has no new pre-cliques over $A\cup\tilde{X}$, any new pre-cliques in $X$ over $A$ are already in $A\cup\tilde{X}$ and hence witnessed by coding nodes in $A$ along possibly with the coding node $c_{*}$ in $\tilde{X}$ (in Case (b)). If this is the case, then the coding node $f(c_{*})$ in $X$ along with those same coding nodes in $A$ witness the new pre-clique in $X$ over $A$. Therefore, $f$ is a strong isomorphism. It follows that since $A\cup\tilde{X}$ has the Witnessing Property, so does $A\cup X$. Also note that if moreover $A\cup\tilde{X}$ has the Strong Witnessing Property, then $A\cup X$ does as well. ∎

In the following, for a finite subtree $A$ of some $T\in\mathcal{T}_{k}$, recall that $\max(A)$ denotes the set $\{t\in A:|t|=l_{A}\}$, the set of all nodes in $A$ of the maximum length, and $A^{+}$ denotes the set of all immediate successors of $\max(A)$ in $\widehat{T}$. We now prove the analogue of the Halpern-Läuchli Theorem for strong coding trees.

Theorem 7.2**.**

Fix $T\in\mathcal{T}_{k}$ and $B$ a finite valid subtree of $T$ such that $B\in\mathcal{AT}^{k}_{m}$, for some $m\geq 1$. Let $A$ be a subtree of $B$ with $l_{A}=l_{B}$ and $0^{(l_{A})}\in A$ such that $A$ is valid in $T$. Let $A_{e}$ be a subset of $A^{+}$ of size at least two such that $0^{(l_{A}+1)}$ is in $A_{e}$. Let $\tilde{X}$ be a level set in $T$ end-extending $A_{e}$ with at least two members, one of which is the node $0^{(l_{\tilde{X}})}$ such that $A\cup\tilde{X}$ is a finite valid subtree of $T$.

Given any coloring $h:\operatorname{Ext}_{T}(A,\tilde{X})\rightarrow 2$, there is a strong coding tree $S\in[B,T]$ such that $h$ is monochromatic on $\operatorname{Ext}_{S}(A,\tilde{X})$. If $\tilde{X}$ has a coding node, then the strong coding tree $S$ is, moreover, taken to be in $[r_{m_{0}-1}(B^{\prime}),T]$, where $m_{0}$ is the integer for which there is a $B^{\prime}\in r_{m_{0}}[B,T]$ with $\tilde{X}\subseteq\max(B^{\prime})$.

Proof.

Let $T,A,A_{e},B,\tilde{X}$ be given satisfying the hypotheses, and let $h$ be a coloring of the members of $\operatorname{Ext}_{T}(A,\tilde{X})$ into two colors, $\{0,1\}$. Fix the following notation: Let $d+1$ equal the number of nodes in $\tilde{X}$, and enumerate the nodes in $\tilde{X}$ as $s_{0},\dots,s_{d}$ so that $s_{d}$ is the critical node in $\tilde{X}$. Let $i_{0}$ denote the integer such that $s_{i_{0}}$ is the node which is a sequence of [math]’s. Notice that $i_{0}$ can equal $d$ only if we are in Case (a) and the splitting node in $\tilde{X}$ is a sequence of [math]’s. In Case (b), let $I_{0}$ denote the set of all $i<d$ such that $s^{+}_{i}(l_{\tilde{X}})=0$ and let $I_{1}$ denote the set of all $i<d$ such that $s^{+}_{i}(l_{\tilde{X}})=1$.

Let $L$ denote the collection of all $l\in\mathbb{N}$ such that there is a member of $\operatorname{Ext}_{T}(A,\tilde{X})$ with nodes of length $l$. In Case (a), since $B$ is valid in $T$, it follows from Lemmas 6.9 and 6.10 that $L$ consists of those $l\in\mathbb{N}$ for which there is a splitting node of length $l$ extending $s_{d}$ and that $L$ is infinite. In Case (b), since $\tilde{X}$ contains a coding node, it follows from Lemma 6.14 that $L$ is exactly the set of all $l\in\mathbb{N}$ for which there is a coding node of length $l$ extending $s_{d}$ and that $L$ is infinite.

For each $i\in(d+1)\setminus\{i_{0}\}$, let $T_{i}=\{t\in T:t\supseteq s_{i}\}$. Let $\mathcal{S}[0]$ denote the set of all sequences of [math]’s of finite length. Let $T_{i_{0}}=\{t\in T:t\supseteq s_{i_{0}}$ and $t\in\mathcal{S}[0]\}$, the collection of all leftmost nodes in $T$ extending $s_{i_{0}}$. Let $\kappa=\beth_{2d}$. The following forcing notion $\mathbb{P}$ adds $\kappa$ many paths through $T_{i}$, for each $i\in d\setminus\{i_{0}\}$, and one path through $T_{d}$. If $i_{0}\neq d$, then $\mathbb{P}$ will add one path through $T_{i_{0}}$, but with $\kappa$ many ordinals labeling this path. We allow this in order to simplify notation.

$\mathbb{P}$ is the set of conditions $p$ such that $p$ is a function of the form

[TABLE]

where $\vec{\delta}_{p}$ is a finite subset of $\kappa$, $l_{p}\in L$, $\{p(i,\delta):\delta\in\vec{\delta}_{p}\}\subseteq T_{i}\restriction l_{p}$ for each $i<d$, and the following hold:

Case (a). (i) $p(d)$ is the splitting node extending $s_{d}$ of length $l_{p}$;

1. (ii)

$\{p(i,\delta):(i,\delta)\in d\times\vec{\delta}_{p}\}\cup\{p(d)\}$ is free in $T$.

Case (b). (i) $p(d)$ is the coding node extending $s_{d}$ of length $l_{p}$;

1. (ii)

For each $\delta\in\vec{\delta}_{p}$, $j\in\{0,1\}$, and $i\in I_{j}$, the passing number of $p(i,\delta)$ at $p(d)$ is $j$.

Given $p\in\mathbb{P}$, the range of $p$ is defined as

[TABLE]

If also $q\in\mathbb{P}$ and $\vec{\delta}_{p}\subseteq\vec{\delta}_{q}$, then we let $\operatorname{ran}(q\restriction\vec{\delta}_{p})$ denote $\{q(i,\delta):(i,\delta)\in d\times\vec{\delta}_{p}\}\cup\{q(d)\}$. In both Cases (a) and (b), the partial ordering on $\mathbb{P}$ is defined as follows: $q\leq p$ if and only if $l_{q}\geq l_{p}$, $\vec{\delta}_{q}\supseteq\vec{\delta}_{p}$, and the following hold:

1. (i)

$q(d)\supseteq p(d)$, and $q(i,\delta)\supseteq p(i,\delta)$ for each $(i,\delta)\in d\times\vec{\delta}_{p}$, and 2. (ii)

$\operatorname{ran}(q\restriction\vec{\delta}_{p})$ has no new pre-cliques over $\operatorname{ran}(p)$.

Since all conditions in $\mathbb{P}$ have ranges which are free in $T$, we shall say that $q$* is valid over $p$* to mean that (ii) holds.

The theorem will be proved in two main parts. In Part I, we check that $\mathbb{P}$ is an atomless partial order and then prove the main Lemma 7.6. In Part II, we apply Lemma 7.6 to build the tree $S$ such that $h$ is monochromatic on $\operatorname{Ext}_{S}(A,\tilde{X})$.

Part I.****

Lemma 7.3**.**

$(\mathbb{P},\leq)$* is an atomless partial ordering.*

Proof.

The order $\leq$ on $\mathbb{P}$ is clearly reflexive and antisymmetric. Transitivity follows from the fact that the requirement (ii) in the definition of the partial order on $\mathbb{P}$ is a transitive property. To see this, suppose that $p\geq q$ and $q\geq r$. Then $\vec{\delta}_{p}\subseteq\vec{\delta}_{q}\subseteq\vec{\delta}_{r}$, $l_{p}\leq l_{q}\leq l_{r}$, $r$ is valid over $q$, and $q$ is valid over $p$. Since $\operatorname{ran}(r\restriction\vec{\delta}_{p})$ is contained in $\operatorname{ran}(r\restriction\vec{\delta}_{q})$ which has no new pre-cliques over $\operatorname{ran}(q)$, it follows that $\operatorname{ran}(r\restriction\vec{\delta}_{p})$ has no new pre-cliques over $\operatorname{ran}(q\restriction\vec{\delta}_{p})$. Since $\operatorname{ran}(q\restriction\vec{\delta}_{p})$ has no new pre-cliques over $\operatorname{ran}(p)$, it follows that $\operatorname{ran}(r\restriction\vec{\delta}_{p})$ has no new pre-cliques over $\operatorname{ran}(p)$. Therefore, $r$ is valid over $p$, so $p\geq r$.

Claim 1**.**

For each $p\in\mathbb{P}$ and $l>l_{p}$, there are $q,r\in\mathbb{P}$ with $l_{q},l_{r}>l$ such that $q,r<p$ and $q$ and $r$ are incompatible.

Proof.

Let $p\in\mathbb{P}$ and $l>l_{p}$ be given, and let $\vec{\delta}$ denote $\vec{\delta}_{p}$ and let $\vec{\delta}_{r}=\vec{\delta}_{q}=\vec{\delta}$. In Case (a), take $q(d)$ and $r(d)$ to be incomparable splitting nodes in $T$ extending $p(d)$ to some lengths greater than $l$. Such splitting nodes exist by Lemma 6.10, showing that strong coding trees are perfect. Let $l_{q}=|q(d)|$ and $l_{r}=|r(d)|$. For each $(i,\delta)\in d\times\vec{\delta}$, let $q(i,\delta)$ be the leftmost extension in $T$ of $p(i,\delta)$ to length $l_{q}$, and let $r(i,\delta)$ be the leftmost extension of $p(i,\delta)$ to length $l_{r}$. Then $q$ and $r$ are members of $\mathbb{P}$. Since $\operatorname{ran}(p)$ is free in $T$, both $\operatorname{ran}(q)$ and $\operatorname{ran}(r)$ are free in $T$ and $\operatorname{ran}(q\restriction\vec{\delta}_{p})$ and $\operatorname{ran}(r\restriction\vec{\delta}_{p})$ have no new pre-cliques over $\operatorname{ran}(p)$, by Lemma 6.9. It follows that $q$ and $r$ are both valid over $p$. Since neither of $q(d)$ and $r(d)$ extends the other, $q$ and $r$ are incompatible.

In Case (b), let $s$ be a splitting node in $T$ of length greater than $l$ extending $p(d)$. Let $k$ be minimal such that $|c^{T}_{k}|\geq|s|$. Let $u,v$ extend $s^{\frown}0,s^{\frown}1$, respectively, leftmost in $T\restriction l^{T}_{k}$. For each $(i,\delta)\in d\times\vec{\delta}_{p}$, let $p^{\prime}(i,\delta)$ be the leftmost extension of $p(i,\delta)$ in $T\restriction l^{T}_{k}$. By Lemma 6.14, there are $q(d)\supseteq u$ and $q(i,\delta)\supseteq p^{\prime}(i,\delta)$, $(i,\delta)\in d\times\vec{\delta}_{p}$, such that

1. (1)

$q(d)$ is a coding node; 2. (2)

$q$ is valid over $p$; 3. (3)

For each $j<2$, $i\in I_{j}$ if and only if the immediate extension of $q(i,\delta)$ is $j$.

Then $q\in\mathbb{P}$ and $q\leq p$. Likewise by Lemma 6.14, there is a condition $r\in\mathbb{P}$ which extends $\{p^{\prime}(i,\delta):(i,\delta)\in d\times\vec{\delta}_{p}\}\cup\{v\}$ such that $r\leq p$. Since the coding nodes $q(d)$ and $r(d)$ are incomparable, $q$ and $r$ are incompatible conditions in $\mathbb{P}$. ∎

It follows from Claim 1 that $\mathbb{P}$ is atomless. ∎

From now on, whenever ambiguity will not arise by doing so, for a condition $p\in\mathbb{P}$, we will use the terminology critical node of $p$ to refer to $p(d)$, which is a splitting node in Case (a) and a coding node in Case (b). Let $\dot{b}_{d}$ be a $\mathbb{P}$-name for the generic path through $T_{d}$; that is, $\dot{b}_{d}=\{\langle p(d),p\rangle:p\in\mathbb{P}\}$. Note that for each $p\in\mathbb{P}$, $p$ forces that $\dot{b}_{d}\restriction l_{p}=p(d)$. By Claim 1, it is dense to force a critical node in $\dot{b}_{d}$ above any given level in $T$, so $\mathbf{1}_{\mathbb{P}}$ forces that the set of levels of critical nodes in $\dot{b}_{d}$ is infinite. Thus, given any generic filter $G$ for $\mathbb{P}$, $\dot{b}_{d}^{G}=\{p(d):p\in G\}$ is a cofinal path of critical nodes in $T_{d}$. Let $\dot{L}_{d}$ be a $\mathbb{P}$-name for the set of lengths of critical nodes in $\dot{b}_{d}$. Note that $\mathbf{1}_{\mathbb{P}}\Vdash\dot{L}_{d}\subseteq L$. Let $\dot{\mathcal{U}}$ be a $\mathbb{P}$-name for a non-principal ultrafilter on $\dot{L}_{d}$. For $i<d$ and $\alpha<\kappa$, let $\dot{b}_{i,\alpha}$ be a $\mathbb{P}$-name for the $\alpha$-th generic branch through $T_{i}$; that is, $\dot{b}_{i,\alpha}=\{\langle p(i,\alpha),p\rangle:p\in\mathbb{P}$ and $\alpha\in\vec{\delta}_{p}\}$. Then for any $p\in\mathbb{P}$ ,

[TABLE]

For $j\in\mathbb{N}$, we let $[\kappa]^{j}$ denote the collection of all $j$-element subsets of $\kappa$. We shall write sets $\{\alpha_{i}:i<d\}$ in $[\kappa]^{d}$ as vectors $\vec{\alpha}=\langle\alpha_{0},\dots,\alpha_{d-1}\rangle$ in strictly increasing order. For $\vec{\alpha}\in[\kappa]^{d}$, we use the following abbreviation:

[TABLE]

Since the branch $\dot{b}_{d}$ is unique, this abbreviation introduces no ambiguity. For any $l<\omega$,

[TABLE]

Using the abbreviations just defined, $h$ is a coloring on sets of nodes of the form $\dot{b}_{\vec{\alpha}}\restriction l$ whenever this is forced to be a member of $\operatorname{Ext}_{T}(A,\tilde{X})$. Given $\vec{\alpha}\in[\kappa]^{d}$ and a condition $p\in\mathbb{P}$ with $\vec{\alpha}\subseteq\vec{\delta}_{p}$, let

[TABLE]

We now set up to prove Lemma 7.6. For each $\vec{\alpha}\in[\kappa]^{d}$, choose a condition $p_{\vec{\alpha}}\in\mathbb{P}$ such that

1. (1)

$\vec{\alpha}\subseteq\vec{\delta}_{p_{\vec{\alpha}}}$. 2. (2)

$X(p_{\vec{\alpha}},\vec{\alpha})\in\operatorname{Ext}_{T}(A,\tilde{X})$. 3. (3)

There is an $\varepsilon_{\vec{\alpha}}\in 2$ such that $p_{\vec{\alpha}}\Vdash$ “$h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon_{\vec{\alpha}}$** for $\dot{\mathcal{U}}$ many $l$ in $\dot{L}_{d}$”.** 4. (4)

$h(X(p_{\vec{\alpha}},\vec{\alpha}))=\varepsilon_{\vec{\alpha}}$.

Properties (1) - (4) can be guaranteed as follows. Recall that $\{s_{i}:i\leq d\}$ enumerates $\tilde{X}$ and that $s_{d}$ is the critical node in $\tilde{X}$. For each $\vec{\alpha}\in[\kappa]^{d}$, define

[TABLE]

Then $p^{0}_{\vec{\alpha}}$ is a condition in $\mathbb{P}$ with $\operatorname{ran}(p^{0}_{\vec{\alpha}})=\tilde{X}$, and $\vec{\delta}_{p_{\vec{\alpha}}^{0}}=\vec{\alpha}$ which implies (1) holds for any $p\leq p^{0}_{\vec{\alpha}}$. The following fact will be used many times.

Claim 2**.**

Given $\vec{\alpha}\in[\kappa]^{d}$, for any $p\leq p_{\vec{\alpha}}^{0}$, the set of nodes $X(p,\vec{\alpha})$ is a member of $\operatorname{Ext}_{T}(A,\tilde{X})$.

Proof.

Suppose $p\leq p_{\vec{\alpha}}^{0}$. Then $p$ is valid over $p_{\vec{\alpha}}^{0}$, so $X(p,\vec{\alpha})$ has no new pre-cliques over $\tilde{X}$. Since $p$ is a condition of $\mathbb{P}$, $X(p,\vec{\alpha})$ is free in $T$ and $A\cup X(p,\vec{\alpha})$ is strongly similar to $A\cup\tilde{X}$. It follows from Lemma 7.1 that $X(p,\vec{\alpha})$ is in $\operatorname{Ext}_{T}(A,\tilde{X})$. ∎

Thus, (2) holds for any $p\leq p_{\vec{\alpha}}^{0}$. Take an extension $p^{1}_{\vec{\alpha}}\leq p^{0}_{\vec{\alpha}}$ which forces $h(\dot{b}_{\vec{\alpha}}\restriction l)$ to be the same value for $\dot{\mathcal{U}}$ many $l\in\dot{L}_{d}$. Since $\mathbb{P}$ is a forcing notion, there is a $p^{2}_{\vec{\alpha}}\leq p_{\vec{\alpha}}^{1}$ deciding a value $\varepsilon_{\vec{\alpha}}$ for which $p^{2}_{\vec{\alpha}}$ forces that $h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon_{\vec{\alpha}}$ for $\dot{\mathcal{U}}$ many $l$ in $\dot{L}_{d}$. Then (3) holds for any $p\leq p_{\vec{\alpha}}^{2}$. If $p_{\vec{\alpha}}^{2}$ satisfies (4), then let $p_{\vec{\alpha}}=p_{\vec{\alpha}}^{2}$. Otherwise, take some $p^{3}_{\vec{\alpha}}\leq p^{2}_{\vec{\alpha}}$ which decides some $l\in\dot{L}_{d}$ such that $l_{p^{2}_{\vec{\alpha}}}<l_{n}^{T}<l\leq l_{p^{3}_{\vec{\alpha}}}$, for some $n$, and $p^{3}_{\vec{\alpha}}$ forces $h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon_{\vec{\alpha}}$. Since $p^{3}_{\vec{\alpha}}$ forces “$\dot{b}_{\vec{\alpha}}\restriction l=\{p^{3}_{\vec{\alpha}}(i,\alpha_{i})\restriction l:i<d\}\cup\{p^{3}_{\vec{\alpha}}(d)\restriction l\}$” and $h$ is defined in the ground model, this means that $p^{3}_{\vec{\alpha}}(d)\restriction l$ is a splitting node in Case (a) and a coding node in Case (b), and

[TABLE]

where $X(p^{3}_{\vec{\alpha}},\vec{\alpha})\restriction l$ denotes $\{p^{3}_{\vec{\alpha}}(i,\alpha_{i})\restriction l:i<d\}\cup\{p^{3}_{\vec{\alpha}}(d)\restriction l\}$. If $l=l_{p^{3}_{\vec{\alpha}}}$, let $p_{\vec{\alpha}}=p_{\vec{\alpha}}^{3}$, and note that $p_{\vec{\alpha}}$ satisfies (1) - (4).

Otherwise, $l<l_{p^{3}_{\vec{\alpha}}}$. In Case (a), let $p_{\vec{\alpha}}$ be defined as follows: Let $\vec{\delta}_{\vec{\alpha}}=\vec{\delta}_{p_{\vec{\alpha}}^{2}}$ and

[TABLE]

Since $p^{3}_{\vec{\alpha}}$ is a condition in $\mathbb{P}$, $\operatorname{ran}(p^{3}_{\vec{\alpha}})$ is free in $T$. Furthermore, $p^{3}_{\vec{\alpha}}\leq p^{2}_{\vec{\alpha}}$ implies that $\operatorname{ran}(p_{\vec{\alpha}}^{3}\restriction\vec{\delta}_{p^{2}_{\vec{\alpha}}})$ has no new pre-cliques over $\operatorname{ran}(p^{2}_{\vec{\alpha}})$. Therefore, leftmost extensions of $\operatorname{ran}(p_{\vec{\alpha}})$ have no new pre-cliques, so $\operatorname{ran}(p_{\vec{\alpha}})$ is free in $T$. Therefore, $p_{\vec{\alpha}}$ is a condition in $\mathbb{P}$ and $p_{\vec{\alpha}}\leq p_{\vec{\alpha}}^{2}$. Thus, $p_{\vec{\alpha}}$ satisfies (1) - (3), and (4) holds by equation (51).

In Case (b), we construct $p_{\vec{\alpha}}\leq p^{2}_{\vec{\alpha}}$ as follows: As in Case (a), let $\vec{\delta}_{\vec{\alpha}}=\vec{\delta}_{p^{2}_{\vec{\alpha}}}$. For each $i<d$, define $p_{\vec{\alpha}}(i,\alpha_{i})=p^{3}_{\vec{\alpha}}(i,\alpha_{i})\restriction l$, and let $p_{\vec{\alpha}}(d)=p^{3}_{\vec{\alpha}}(d)\restriction l$. Then $X(p_{\vec{\alpha}},\vec{\alpha})=\{p^{3}_{\vec{\alpha}}(i,\alpha_{i})\restriction l:i<d\}\cup\{p^{3}_{\vec{\alpha}}(d)\restriction l\}$, so $h(X(p_{\vec{\alpha}},\vec{\alpha}))=\varepsilon_{\vec{\alpha}}$. Let $U$ denote $X(p_{\vec{\alpha}}^{2},\vec{\alpha})$ and let $U^{\prime}=\operatorname{ran}(p_{\vec{\alpha}}^{2})\setminus U$. Let $X$ denote $X(p_{\vec{\alpha}},\vec{\alpha})$ and note that $X$ end-extends $U$, and $X$ is free in $T$ and has no new pre-cliques over $U$. By Lemma 6.17, there is an $X^{\prime}$ end-extending $U^{\prime}$ to nodes in $T\restriction l$ so that the following hold: $X\cup X^{\prime}$ is free in $T$ and has no new pre-cliques over $U\cup U^{\prime}$; furthermore, each node in $X^{\prime}$ has the same passing number at $l$ as it does at $l_{p_{\vec{\alpha}}^{2}}$. Let $\operatorname{ran}(p_{\vec{\alpha}})$ be this set of nodes $X\cup X^{\prime}$, where for each $i<d$ and $(i,\delta)\in d\times\vec{\delta}_{p_{\vec{\alpha}}^{3}}$ with $\delta\neq\alpha_{i}$, we let $p_{\vec{\alpha}}(i,\delta)$ be the node in $Y^{\prime}$ extending $p_{\vec{\alpha}}^{3}(i,\delta)$. This defines a condition $p_{\vec{\alpha}}\leq p_{\vec{\alpha}}^{2}$ satisfying (1) - (4).

The rest of Part I follows by arguments in ****[14]**** for the case $k=3$, with no modifications. It is included here for the reader’s convenience. We are assuming $\kappa=\beth_{2d}$ so that $\kappa\rightarrow(\aleph_{1})^{2d}_{\aleph_{0}}$, by the Erdős-Rado Theorem (Theorem 3.9). Given two sets of ordinals $J,K$ we shall write $J<K$ if every member of $J$ is less than every member of $K$. Let $D_{e}=\{0,2,\dots,2d-2\}$ and $D_{o}=\{1,3,\dots,2d-1\}$, the sets of even and odd integers less than $2d$, respectively. Let $\mathcal{I}$ denote the collection of all functions $\iota:2d\rightarrow 2d$ such that $\iota\restriction D_{e}$ and $\iota\restriction D_{o}$ are strictly increasing sequences and $\{\iota(0),\iota(1)\}<\{\iota(2),\iota(3)\}<\dots<\{\iota(2d-2),\iota(2d-1)\}$. Thus, each $\iota$ codes two strictly increasing sequences $\iota\restriction D_{e}$ and $\iota\restriction D_{o}$, each of length $d$. For $\vec{\theta}\in[\kappa]^{2d}$, $\iota(\vec{\theta}\,)$ determines the pair of sequences of ordinals $(\theta_{\iota(0)},\theta_{\iota(2)},\dots,\theta_{\iota(2d-2))}),(\theta_{\iota(1)},\theta_{\iota(3)},\dots,\theta_{\iota(2d-1)})$, both of which are members of $[\kappa]^{d}$. Denote these as $\iota_{e}(\vec{\theta}\,)$ and $\iota_{o}(\vec{\theta}\,)$, respectively. To ease notation, let $\vec{\delta}_{\vec{\alpha}}$ denote $\vec{\delta}_{p_{\vec{\alpha}}}$, $k_{\vec{\alpha}}$ denote $|\vec{\delta}_{\vec{\alpha}}|$, and let $l_{\vec{\alpha}}$ denote $l_{p_{\vec{\alpha}}}$. Let $\langle\delta_{\vec{\alpha}}(j):j<k_{\vec{\alpha}}\rangle$ denote the enumeration of $\vec{\delta}_{\vec{\alpha}}$ in increasing order.

Define a coloring $f$ on $[\kappa]^{2d}$ into countably many colors as follows: Given $\vec{\theta}\in[\kappa]^{2d}$ and $\iota\in\mathcal{I}$, to reduce the number of subscripts, letting $\vec{\alpha}$ denote $\iota_{e}(\vec{\theta}\,)$ and $\vec{\beta}$ denote $\iota_{o}(\vec{\theta}\,)$, define

[TABLE]

Let $f(\vec{\theta}\,)$ be the sequence $\langle f(\iota,\vec{\theta}\,):\iota\in\mathcal{I}\rangle$, where $\mathcal{I}$ is given some fixed ordering. Since the range of $f$ is countable, apply the Erdős-Rado Theorem to obtain a subset $K\subseteq\kappa$ of cardinality $\aleph_{1}$ which is homogeneous for $f$. Take $K^{\prime}\subseteq K$ such that between each two members of $K^{\prime}$ there is a member of $K$. Take subsets $K_{i}\subseteq K^{\prime}$ such that $K_{0}<\dots<K_{d-1}$ and each $|K_{i}|=\aleph_{0}$.

Lemma 7.4**.**

There are $\varepsilon^{*}\in 2$, $k^{*}\in\omega$, $t_{d}$, and $\langle t_{i,j}:j<k^{*}\rangle$, $i<d$, such that for all $\vec{\alpha}\in\prod_{i<d}K_{i}$ and each $i<d$, $\varepsilon_{\vec{\alpha}}=\varepsilon^{*}$, $k_{\vec{\alpha}}=k^{*}$, $p_{\vec{\alpha}}(d)=t_{d}$, and $\langle p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j)):j<k_{\vec{\alpha}}\rangle=\langle t_{i,j}:j<k^{*}\rangle$.

Proof.

Let $\iota$ be the member in $\mathcal{I}$ which is the identity function on $2d$. For any pair $\vec{\alpha},\vec{\beta}\in\prod_{i<d}K_{i}$, there are $\vec{\theta},\vec{\theta}^{\prime}\in[K]^{2d}$ such that $\vec{\alpha}=\iota_{e}(\vec{\theta}\,)$ and $\vec{\beta}=\iota_{e}(\vec{\theta}^{\prime}\,)$. Since $f(\iota,\vec{\theta}\,)=f(\iota,\vec{\theta}^{\prime}\,)$, it follows that $\varepsilon_{\vec{\alpha}}=\varepsilon_{\vec{\beta}}$, $k_{\vec{\alpha}}=k_{\vec{\beta}}$, $p_{\vec{\alpha}}(d)=p_{\vec{\beta}}(d)$, and $\langle\langle p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j)):j<k_{\vec{\alpha}}\rangle:i<d\rangle=\langle\langle p_{\vec{\beta}}(i,\delta_{\vec{\beta}}(j)):j<k_{\vec{\beta}}\rangle:i<d\rangle$. Thus, define $\varepsilon^{*}$, $k^{*}$, $t_{d}$, $\langle\langle t_{i,j}:j<k^{*}\rangle:i<d\rangle$ to be $\varepsilon_{\vec{\alpha}}$, $k_{\vec{\alpha}}$, $p_{\vec{\alpha}}(d)$, $\langle\langle p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j)):j<k_{\vec{\alpha}}\rangle:i<d\rangle$ for any $\vec{\alpha}\in\prod_{i<d}K_{i}$. ∎

Let $l^{*}$ denote the length of $t_{d}$. Then all the nodes $t_{i,j}$, $i<d$, $j<k^{*}$, also have length $l^{*}$.

Lemma 7.5**.**

Given any $\vec{\alpha},\vec{\beta}\in\prod_{i<d}K_{i}$, if $j,k<k^{*}$ and $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(k)$, then $j=k$.

Proof.

Let $\vec{\alpha},\vec{\beta}$ be members of $\prod_{i<d}K_{i}$ and suppose that $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(k)$ for some $j,k<k^{*}$. For each $i<d$, let $\rho_{i}$ be the relation from among $\{<,=,>\}$ such that $\alpha_{i}\,\rho_{i}\,\beta_{i}$. Let $\iota$ be the member of $\mathcal{I}$ such that for each $\vec{\gamma}\in[K]^{d}$ and each $i<d$, $\theta_{\iota(2i)}\ \rho_{i}\ \theta_{\iota(2i+1)}$. Then there is a $\vec{\theta}\in[K^{\prime}]^{2d}$ such that $\iota_{e}(\vec{\theta})=\vec{\alpha}$ and $\iota_{o}(\vec{\theta})=\vec{\beta}$. Since between any two members of $K^{\prime}$ there is a member of $K$, there is a $\vec{\gamma}\in[K]^{d}$ such that for each $i<d$, $\alpha_{i}\,\rho_{i}\,\gamma_{i}$ and $\gamma_{i}\,\rho_{i}\,\beta_{i}$, and furthermore, for each $i<d-1$, $\{\alpha_{i},\beta_{i},\gamma_{i}\}<\{\alpha_{i+1},\beta_{i+1},\gamma_{i+1}\}$. Given that $\alpha_{i}\,\rho_{i}\,\gamma_{i}$ and $\gamma_{i}\,\rho_{i}\,\beta_{i}$ for each $i<d$, there are $\vec{\mu},\vec{\nu}\in[K]^{2d}$ such that $\iota_{e}(\vec{\mu})=\vec{\alpha}$, $\iota_{o}(\vec{\mu})=\vec{\gamma}$, $\iota_{e}(\vec{\nu})=\vec{\gamma}$, and $\iota_{o}(\vec{\nu})=\vec{\beta}$. Since $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(k)$, the pair $\langle j,k\rangle$ is in the last sequence in $f(\iota,\vec{\theta})$. Since $f(\iota,\vec{\mu})=f(\iota,\vec{\nu})=f(\iota,\vec{\theta})$, also $\langle j,k\rangle$ is in the last sequence in $f(\iota,\vec{\mu})$ and $f(\iota,\vec{\nu})$. It follows that $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\gamma}}(k)$ and $\delta_{\vec{\gamma}}(j)=\delta_{\vec{\beta}}(k)$. Hence, $\delta_{\vec{\gamma}}(j)=\delta_{\vec{\gamma}}(k)$, and therefore $j$ must equal $k$. ∎

For any $\vec{\alpha}\in\prod_{i<d}K_{i}$ and any $\iota\in\mathcal{I}$, there is a $\vec{\theta}\in[K]^{2d}$ such that $\vec{\alpha}=\iota_{o}(\vec{\theta})$. By homogeneity of $f$ and by the first sequence in the second line of equation (53), there is a strictly increasing sequence $\langle j_{i}:i<d\rangle$ of members of $k^{*}$ such that for each $\vec{\alpha}\in\prod_{i<d}K_{i}$, $\delta_{\vec{\alpha}}(j_{i})=\alpha_{i}$. For each $i<d$, let $t^{*}_{i}$ denote $t_{i,j_{i}}$. Then for each $i<d$ and each $\vec{\alpha}\in\prod_{i<d}K_{i}$,

[TABLE]

Let $t_{d}^{*}$ denote $t_{d}$.

Lemma 7.6**.**

For any finite subset $\vec{J}\subseteq\prod_{i<d}K_{i}$, the set of conditions $\{p_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\,\}$ is compatible. Moreover, $p_{\vec{J}}:=\bigcup\{p_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\,\}$ is a member of $\mathbb{P}$ which is below each $p_{\vec{\alpha}}$, $\vec{\alpha}\in\vec{J}$.

Proof.

For any $\vec{\alpha},\vec{\beta}\in\prod_{i<d}K_{i}$, whenver $j,k<k^{*}$ and $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(k)$, then $j=k$, by Lemma 7.5. It then follows from Lemma 7.4 that for each $i<d$,

[TABLE]

Thus, for each $\vec{\alpha},\vec{\beta}\in\vec{J}$ and each $\delta\in\vec{\delta}_{\vec{\alpha}}\cap\vec{\delta}_{\vec{\beta}}$, for all $i<d$,

[TABLE]

Thus, $p_{\vec{J}}:=\bigcup\{p_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\}$ is a function. Let $\vec{\delta}_{\vec{J}}=\bigcup\{\vec{\delta}_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\,\}$. For each $\delta\in\vec{\delta}_{\vec{J}}$ and $i<d$, $p_{\vec{J}}(i,\delta)$ is defined, and it is exactly $p_{\vec{\alpha}}(i,\delta)$, for any $\vec{\alpha}\in\vec{J}$ such that $\delta\in\vec{\delta}_{\vec{\alpha}}$. Thus, $p_{\vec{J}}$ is a member of $\mathbb{P}$, and $p_{\vec{J}}\leq p_{\vec{\alpha}}$ for each $\vec{\alpha}\in\vec{J}$. ∎

The final lemma of Part I will be used in the next section.

Lemma 7.7**.**

If $\beta\in\bigcup_{i<d}K_{i}$, $\vec{\alpha}\in\prod_{i<d}K_{i}$, and $\beta\not\in\vec{\alpha}$, then $\beta$ is not a member of $\vec{\delta}_{\vec{\alpha}}$.

Proof.

Suppose toward a contradiction that $\beta\in\vec{\delta}_{\vec{\alpha}}$. Then there is a $j<k^{*}$ such that $\beta=\delta_{\vec{\alpha}}(j)$. Let $i$ be such that $\beta\in K_{i}$. Since $\beta\neq\alpha_{i}=\delta_{\vec{\alpha}}(j_{i})$, it must be that $j\neq j_{i}$. However, letting $\vec{\beta}$ be any member of $\prod_{i<d}K_{i}$ with $\beta_{i}=\beta$, then $\beta=\delta_{\vec{\beta}}(j_{i})=\delta_{\vec{\alpha}}(j)$, so Lemma 7.5 implies that $j_{i}=j$, a contradiction. ∎

Part II.** In this last part of the proof, we build a strong coding tree $S$ valid in $T$ on which the coloring $h$ is homogeneous. Cases (a) and (b) must be handled separately.**

Part II Case (a).** Recall that $\{s_{i}:i\leq d\}$ enumerates the members of $A_{e}$, which is a subset of $B^{+}$. Let $m$ be the integer such that $B\in\mathcal{AT}_{m}^{k}$. Let $M=\{m_{j}:j\in\mathbb{N}\}$ be the strictly increasing enumeration of those $m^{\prime}>m$ such that the splitting node in $\max(r_{m^{\prime}}(T))$ extends $s_{d}$. By induction on $j$ we will construct the following: The base case splits into two subcases depending on whether $m_{0}>m+1$ or $m_{0}=m+1$. In the first case, we will find some $U_{m_{0}-1}\in r_{m_{0}-1}[B,T]$ which is valid in $T$. In the second, we will find some $U_{m_{1}-1}\in r_{m_{1}-1}[B,T]$ which is valid in $T$ and such that $h$ takes color $\varepsilon^{*}$ on $\operatorname{Ext}_{U_{m_{1}-1}}(A,\tilde{X})$. In general, given $U_{m_{j}-1}$, we will use the forcing to find some $U_{m_{j}}\in r_{m_{j}}[U_{m_{j}-1},T]$ which is valid in $T$ such that $h$ takes color $\varepsilon^{*}$ on $\operatorname{Ext}_{U_{m_{j}}}(A,\tilde{X})$. Then we will apply Theorem 6.16 and Lemma 6.18 to find an extension $U_{m_{j+1}-1}\in r_{m_{j+1}-1}[U_{m_{j}},T]$ which is valid in $T$ and continue the induction. Setting $S=\bigcup_{j\in\mathbb{N}}U_{m_{j}}$ will yield $S$ to be a member of $[B,T]$ for which $\operatorname{Ext}_{S}(A,\tilde{X})$ is homogeneous for $h$, with color $\varepsilon^{*}$.**

First extend each node in $B^{+}$ to level $l^{*}$ as follows. The set $\{t^{*}_{i}:i\leq d\}$ end-extends $A_{e}$, has no new pre-cliques over $A_{e}$, and is free in $T$. For each node $u$ in $B^{+}\setminus A_{e}$, let $u^{*}$ denote its leftmost extension in $T\restriction l^{*}$. Then the set

[TABLE]

end-extends $B^{+}$, is free in $T$, and has no new pre-cliques over $B$, by Lemma 6.9. Thus, $U^{*}$ is free in $T$, and $B\cup U^{*}$ satisfies the Witnessing Property so is valid in $T$. If $m_{0}>m+1$, apply Lemma 6.18 and Theorem 6.16 to extend above $U^{*}$ to construct a member $U_{m_{0}-1}\in r_{m_{0}-1}[B,T]$ which is valid in $T$. In this case, note that $\max(r_{m+1}(U_{m_{0}}))$ is not $U^{*}$, but rather $\max(r_{m+1}(U_{m_{0}}))$ end-extends $U^{*}$.

If $m_{0}=m+1$, then $B\cup U^{*}$ is a member of $r_{m_{0}}[B,T]$, by Lemma 6.18. In this case, we can simply let $U_{m_{0}}=B\cup U^{*}$. Then $h$ takes color $\varepsilon^{*}$ on $\operatorname{Ext}_{U_{m_{0}}}(A,\tilde{X})$, since $U^{*}$ is the only member of that set. Using Theorem 6.16, extend $U_{m_{0}}$ to a member $U_{m_{1}-1}\in r_{m_{1}-1}[U_{m_{0}},T]$ which is valid in $T$.

Assume $j<\omega$ and we have constructed $U_{m_{j}-1}$, valid in $T$, so that every member of $\operatorname{Ext}_{U_{m_{j}-1}}(A,\tilde{X})$ is colored $\varepsilon^{*}$ by $h$. Fix some $C\in r_{m_{j}}[U_{m_{j}-1},T]$ with $C$ valid in $T$, and let $Z=\max(C)$. The nodes in $Z$ will not be in the tree $S$ we are constructing; rather, we will extend the nodes in $Z$ to construct $U_{m_{j}}\in r_{m_{j}}[U_{m_{j}-1},T]$.

We now start to construct a condition $q$ which will satisfy Lemma 7.8, below. Let $q(d)$ denote the splitting node in $Z$ and let $l_{q}=|q(d)|$. For each $i<d$, let $Z_{i}$ denote the set of those $z\in T_{i}\cap Z$ such that $z\in X$ for some $X\in\operatorname{Ext}_{Z}(A,\tilde{X})$. For each $i<d$, take a set $J_{i}\subseteq K_{i}$ of cardinality $|Z_{i}|$ and label the members of $Z_{i}$ as $\{z_{\alpha}:\alpha\in J_{i}\}$. Notice that each member of $\operatorname{Ext}_{T}(A,\tilde{X})$ above $Z$ extends some set $\{z_{\alpha_{i}}:i<d\}\cup\{q(d)\}$, where each $\alpha_{i}\in J_{i}$. Let $\vec{J}$ denote the set of those $\langle\alpha_{0},\dots,\alpha_{d-1}\rangle\in\prod_{i<d}J_{i}$ such that the set $\{z_{\alpha_{i}}:i<d\}\cup\{q(d)\}$ is in $\operatorname{Ext}_{T}(A,\tilde{X})$. Then for each $i<d$, $J_{i}=\{\alpha_{i}:\vec{\alpha}\in\vec{J}\}$. It follows from Lemma 7.6 that the set $\{p_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\}$ is compatible. The fact that $p_{\vec{J}}$ is a condition in $\mathbb{P}$ will be used to make the construction of $q$ very precise.

Let $\vec{\delta}_{q}=\bigcup\{\vec{\delta}_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\}$. For each $i<d$ and $\alpha\in J_{i}$, define $q(i,\alpha)=z_{\alpha}$. Notice that for each $\vec{\alpha}\in\vec{J}$ and $i<d$,

[TABLE]

and

[TABLE]

For each $i<d$ and $\gamma\in\vec{\delta}_{q}\setminus J_{i}$, there is at least one $\vec{\alpha}\in\vec{J}$ and some $k<k^{*}$ such that $\delta_{\vec{\alpha}}(k)=\gamma$. Let $q(i,\gamma)$ be the leftmost extension of $p_{\vec{J}}(i,\gamma)$ in $T$ of length $l_{q}$. Define

[TABLE]

Since $C$ is valid in $T$ and $Z=\max(C)$, it follows that $Z$ is free in $T$. Since $\operatorname{ran}(q)$ consists of $Z$ along with leftmost extensions of nodes in $\operatorname{ran}(p_{\vec{J}}(i,\gamma))$, all of which are free, $\operatorname{ran}(q)$ is free. Therefore, $q$ is a condition in $\mathbb{P}$.

Lemma 7.8**.**

For all $\vec{\alpha}\in\vec{J}$, $q\leq p_{\vec{\alpha}}$.

Proof.

Given $\vec{\alpha}\in\vec{J}$, it follows from the definition of $q$ that $\vec{\delta}_{q}\supseteq\vec{\delta}_{\vec{\alpha}}$, $q(d)\supseteq p_{\vec{\alpha}}(d)$, and for each pair $(i,\gamma)\in d\times\vec{\delta}_{\vec{\alpha}}$, $q(i,\gamma)\supseteq p_{\vec{\alpha}}(i,\gamma)$. So it only remains to show that $q$ is valid over $p_{\vec{\alpha}}$. It follows from Lemma 7.7 that $\vec{\delta}_{\vec{\alpha}}\cap\bigcup_{i<d}K_{i}=\vec{\alpha}$; so for each $i<d$ and $\gamma\in\vec{\delta}_{\vec{\alpha}}\setminus\{\alpha_{i}\}$, $q(i,\gamma)$ is the leftmost extension of $p_{\vec{\alpha}}(i,\gamma)$. Since $\vec{\alpha}$ is in $\vec{J}$, $X(q,\vec{\alpha})$ is in $\operatorname{Ext}_{T}(A,\tilde{X})$. This implies that $X(q,\vec{\alpha})$ has no new pre-cliques over $A$, and hence, none over $X(p_{\vec{\alpha}},\vec{\alpha})$. It follows that $\operatorname{ran}(q\restriction\vec{\delta}_{\vec{\alpha}})$ is valid over $\operatorname{ran}(p_{\vec{\alpha}})$, by Lemma 6.9. Therefore, $q\leq p_{\vec{\alpha}}$. ∎

Remark 7.9*.*

Notice that we did not prove that $q\leq p_{\vec{J}}$; in fact that is generally false.

To construct $U_{m_{j}}$, take an $r\leq q$ in $\mathbb{P}$ which decides some $l_{j}$ in $\dot{L}_{d}$ for which $h(\dot{b}_{\vec{\alpha}}\restriction l_{j})=\varepsilon^{*}$, for all $\vec{\alpha}\in\vec{J}$. This is possible since for all $\vec{\alpha}\in\vec{J}$, $p_{\vec{\alpha}}$ forces $h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon^{*}$ for $\dot{\mathcal{U}}$ many $l\in\dot{L}_{d}$. By the same argument as in creating the conditions $p_{\vec{\alpha}}\leq p_{\vec{\alpha}}^{2}$ to satisfy (4) in Part I, we may assume that the nodes in the image of $r$ have length $l_{j}$. Since $r$ forces $\dot{b}_{\vec{\alpha}}\restriction l_{j}=X(r,\vec{\alpha})$ for each $\vec{\alpha}\in\vec{J}$, and since the coloring $h$ is defined in the ground model, it follows that $h(X(r,\vec{\alpha}))=\varepsilon^{*}$ for each $\vec{\alpha}\in\vec{J}$. Extend the splitting node $q(d)$ in $Z$ to $r(d)$. For each $i<d$ and $\alpha_{i}\in J_{i}$, extend $q(i,\alpha_{i})$ to $r(i,\alpha_{i})$. Let

[TABLE]

and let $Z_{1}=Z\setminus Z_{0}$. Let

[TABLE]

Then $Y$ extends $Z_{0}$ and has no new pre-cliques over $Z_{0}$, since $r\leq q$. By Lemma 6.18, there is a $U_{m_{j}}\in r_{m_{j}}[U_{m_{j}-1},T]$ which is valid in $T$ such that $\max(U_{m_{j}})$ end-extends $Z$ and in particular, $Y\subseteq\max(U_{m_{j}})$. Notice that every $X\in\operatorname{Ext}_{U_{m_{j}}}(A,\tilde{X})$ with $X\subseteq\max(U_{m_{j}})$ satisfies $h(X)=\varepsilon^{*}$. This holds since for each such $X$, the truncation $X\restriction l_{q}$ is a member of $\operatorname{Ext}_{Z}(A,\tilde{X})$. So there corresponds a sequence $\vec{\alpha}\in\vec{J}$ such that $X\restriction l_{q}=X(q,\vec{\alpha})$. Then $X=X(r,\vec{\alpha})$, which has $h$-color $\varepsilon^{*}$.

To finish the induction step, apply Lemma 6.18 and Theorem 6.16 to extend $U_{m_{j}}$ to some $U_{m_{j+1}-1}\in r_{m_{j+1}-1}[U_{m_{j}},T]$ which is valid in $T$. Note that every member of $\operatorname{Ext}_{U_{m_{j+1}-1}}(A,\tilde{X})$ is colored $\varepsilon^{*}$ by $h$, since $\operatorname{Ext}_{U_{m_{j+1}-1}}(A,\tilde{X})=\operatorname{Ext}_{U_{m_{j}}}(A,\tilde{X})$.

Let $S=\bigcup_{j\in\mathbb{N}}U_{m_{j}}$. Then for each $X\in\operatorname{Ext}_{S}(A,\tilde{X})$, there corresponds a $j$ such that $X\in\operatorname{Ext}_{U_{m_{j}}}(A,\tilde{X})$, and hence, $h(X)=\varepsilon^{*}$. Thus, $S\in[B,T]$ and satisfies the theorem. This concludes the proof of the theorem for Case (a).

Part II Case (b).** Let $m_{0}$ be the integer such that there is a $B^{\prime}\in r_{m_{0}}[B,T]$ with $\tilde{X}\subseteq\max(B^{\prime})$. Let $U_{m_{0}-1}$ denote $r_{m_{0}-1}(B^{\prime})$. Since $\tilde{X}\subseteq\max(B^{\prime})$, it follows that $l^{*}\geq l_{B^{\prime}}$. Let $V=\{t^{*}_{i}:i\leq d\}$, and recall that this set has no new pre-cliques over $\tilde{X}$. By Lemma 6.18 there is a set of nodes $V^{\prime}$ end-extending $\max(B^{\prime})\setminus V$ such that $U_{m_{0}-1}\cup V\cup V^{\prime}$ is a member of $r_{m_{0}}[U_{m_{0}-1},T]$; label this $U_{m_{0}}$. Since $\max(U_{m_{0}})$ is at the level of the coding node $t^{*}_{d}$, $\max(U_{m_{0}})$ is free in $T$. Moreover, $U_{m_{0}}\in r_{m_{0}}[U_{m_{0}-1},T]$ implies that $U_{m_{0}}$ satisfies the Strong Witnessing Property. Therefore, $U_{m_{0}}$ is valid in $T$. Notice that $\{t^{*}_{i}:i\leq d\}$ is the only member of $\operatorname{Ext}_{U_{m_{0}}}(A,\tilde{X})$, and it has $h$-color $\varepsilon^{*}$.**

Let $M=\{m_{j}:j\in\mathbb{N}\}$ enumerate the set of $m\geq m_{0}$ such that the coding node $c^{T}_{m}\supseteq c^{T}_{m_{0}}$. Assume that $j\geq 1$ and we have constructed $U_{m_{j-1}}\in\mathcal{AT}^{k}_{m_{j-1}}$ valid in $T$ so that every member of $\operatorname{Ext}_{U_{m_{j-1}}}(A,\tilde{X})$ is colored $\varepsilon^{*}$ by $h$. By Theorem 6.16, we may fix some $U_{m_{j}-1}\in r_{m_{j}-1}[U_{m_{j-1}},T]$ which is valid in $T$. Take some $C\in r_{m_{j}}[U_{m_{j}-1},T]$, and let $Z$ denote $\max(C)$. The nodes in $Z$ will not be in the tree $S$ we are constructing; rather, we will construct $U_{m_{j}}\in r_{m_{j}}[U_{m_{j}-1},T]$ so that $\max(U_{m_{j}})$ extends $Z$. Let $q(d)$ denote the coding node in $Z$ and let $l_{q}=|q(d)|$. Recall that for $e\in\{0,1\}$, $I_{e}$ denotes the set of $i<d$ for which $t^{*}_{i}$ has passing number $e$ at $t^{*}_{d}$. For each pair $e\in\{0,1\}$ and $i\in I_{e}$, let $Z_{i}$ be the set of nodes $z$ in $T_{i}\cap Z$ such that $z$ has passing number $e$ at $q(d)$.

We now construct a condition $q$ similarly to, but not exactly as in, Case (a). For each $i<d$, let $J_{i}$ be a subset of $K_{i}$ with the same size as $Z_{i}$. For each $i<d$, label the nodes in $Z_{i}$ as $\{z_{\alpha}:\alpha\in J_{i}\}$. Let $\vec{J}$ denote the set of those $\langle\alpha_{0},\dots,\alpha_{d-1}\rangle\in\prod_{i<d}J_{i}$ such that the set $\{z_{\alpha_{i}}:i<d\}\cup\{q(d)\}$ is in $\operatorname{Ext}_{T}(A,\tilde{X})$. Notice that for each $i<d$ and $\vec{\alpha}\in\vec{J}$, $z_{\alpha_{i}}\supseteq t^{*}_{i}=p_{\vec{\alpha}}(i,\alpha_{i})$, and $q(d)\supseteq t^{*}_{d}=p_{\vec{\alpha}}(d)$. Furthermore, for each $i<d$ and $\delta\in J_{i}$, there is an $\vec{\alpha}\in\vec{J}$ such that $\alpha_{i}=\delta$. Let $\vec{\delta}_{q}=\bigcup\{\vec{\delta}_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\,\}$. For each pair $(i,\gamma)\in d\times\vec{\delta}_{q}$ with $\gamma\in J_{i}$, define $q(i,\gamma)=z_{\gamma}$.

Let $\mathcal{J}=\{(i,\gamma)\in d\times\vec{\delta}_{q}:i<d$ and $\gamma\in\vec{\delta}_{q}\setminus J_{i}\}$. For each pair $(i,\gamma)\in\mathcal{J}$, there is at least one $\vec{\alpha}\in\vec{J}$ and some $k<k^{*}$ such that $\delta_{\vec{\alpha}}(k)=\gamma$. By Lemma 7.6, $p_{\vec{\beta}}(i,\gamma)=p_{\vec{\alpha}}(i,\gamma)=t^{*}_{i,k}$, for any $\vec{\beta}\in\vec{J}$ for which $\gamma\in\vec{\delta}_{\vec{\beta}}$. For each pair $(i,\gamma)\in\mathcal{J}$ with $i\in I_{0}$, take $q(i,\gamma)$ to be the leftmost extension of $t^{*}_{i,k}$ in $T\restriction l_{q}$. For each pair $(i,\gamma)\in\mathcal{J}$ with $i\in I_{1}$, let $q(i,\gamma)$ be the node which extends $t^{*}_{i,k}$ leftmost until length of the longest coding node in $T$ strictly below $q(d)$, and then takes the rightmost path to length $l_{q}$. Note that $q(i,\gamma)$ has passing number $e$, where $e\in\{0,1\}$ is the number such that $i\in I_{e}$. By similar arguments to those in Lemma 6.17, the set $\{q(i,\gamma):(i,\gamma)\in\mathcal{J}\}$ has no new pre-cliques over $\operatorname{ran}(p_{\vec{\alpha}})$ for $\vec{\alpha}\in\vec{J}$ (recall, these all have the same range); moreover, any new pre-cliques in the set $\{q(i,\gamma):i<d,\ \gamma\in\vec{\delta}_{q}\}\cup\{q(d)\}$ over $\operatorname{ran}(p_{\vec{\alpha}})$ (for any $\vec{\alpha}\in\vec{J}$) must occur among $\{q(i,\gamma):i<d,\ \gamma\in J_{i}\}\cup\{q(d)\}$.

Define

[TABLE]

By the construction, $q$ is a member of $\mathbb{P}$.

Claim 3**.**

For each $\vec{\alpha}\in\vec{J}$, $q\leq p_{\vec{\alpha}}$.

Proof.

By construction, $q(i,\delta)\supseteq p_{\vec{\alpha}}(i,\delta)$ for all $(i,\delta)\in d\times\vec{\delta}_{\vec{\alpha}}$; so it suffices to show that for each $\vec{\alpha}\in\vec{J}$, $\operatorname{ran}(q\restriction\vec{\delta}_{\vec{\alpha}})$ has no new pre-cliques over $\operatorname{ran}(p_{\vec{\alpha}})$.

Let $\vec{\alpha}\in\vec{J}$ be given. Then

[TABLE]

recalling that $X(q,\vec{\alpha})=\{q(i,\alpha_{i}):i<d\}\cup\{q(d)\}$. By definition of $\vec{J}$, $\vec{\alpha}\in\vec{J}$ implies that $X(q,\vec{\alpha})$ is a member of $\operatorname{Ext}_{T}(A,\tilde{X})$. Thus, $X(q,\vec{\alpha})$ has no new pre-cliques over $A\cup\tilde{X}$, by Lemma 7.1. Since $\{t^{*}_{i}:i\leq d\}$ end-extends $\tilde{X}$, it follows that $X(q,\vec{\alpha})$ has no new pre-cliques over $\{t^{*}_{i}:i\leq d\}$. Since the set $\{q(i,\gamma):i\leq d,\ \gamma\in\vec{\delta}_{\vec{\alpha}},\ \gamma\neq\alpha_{i}\}$ has no new pre-cliques with $X(q,\vec{\alpha})$ over $\operatorname{ran}(p_{\vec{\alpha}})$, it follows that $q\leq p_{\vec{\alpha}}$. ∎

To construct $U_{m_{j}}$, take an $r\leq q$ in $\mathbb{P}$ which decides $l_{r}\in\dot{L}_{d}$ such that $h(\dot{b}_{\vec{\alpha}}\restriction l_{r})=\varepsilon^{*}$ for all $\vec{\alpha}\in\vec{J}$, using the same ideas as in the construction of the $p_{\vec{\alpha}}$’s. Let $Y=\bigcup\{X(r,\vec{\alpha}):\vec{\alpha}\in\vec{J}\}$, and let $Z^{*}=\{r(d)\}\cup\bigcup_{i<d}Z_{i}$. Since $\operatorname{ran}(r\restriction\vec{\delta}_{q})$ has no new pre-cliques over $\operatorname{ran}(q)$, it follows that $Y$ has no new pre-cliques over $Z^{*}$. Apply Lemma 6.17 to extend the nodes in $Z\setminus Z^{*}$ to a set $Y^{\prime}\subseteq T\restriction l_{r}$ so that each node in $Y^{\prime}$ has the same passing number at $r(d)$ as it does at $q(d)$, and such that $Y\cup Y^{\prime}$ has no new pre-cliques over $Z$. Then $U_{m_{j}-1}\cup Y\cup Y^{\prime}$ is a member of $r_{m_{j}}[U_{m_{j}-1},T]$ which is valid in $T$.

To finish the proof of the theorem for Case (b), Define $S=\bigcup_{j\in\mathbb{N}}U_{m_{j}}$. Then $S\in[B^{\prime},T]$, and for each $Z\in\operatorname{Ext}_{S}(A,\tilde{X})$, there is a $j\in\mathbb{N}$ such that $Z\in\operatorname{Ext}_{U_{m_{j}}}(A,\tilde{X})$, so $h(Z)=\varepsilon^{*}$.

This concludes the proof of the theorem. ∎

8. Ramsey Theorem for finite trees with the Strict Witnessing Property

The main theorem of this section is Theorem 8.3, which is an analogue of Milliken’s Theorem 3.5 for colorings of finite trees with the following strong version of the Witnessing Property.

Definition 8.1** (Strict Witnessing Property).**

A subtree $A$ of a strong coding tree satisfies the Strict Witnessing Property (SWP) if $A$ satisfies the Witnessing Property and the following hold:

1. (1)

For each interval $(|d_{m}^{A}|,|d^{A}_{m+1}|]$, $A$ has at most one new pre-clique of size at least two, or a singleton with some new pre-cliques in $(|d_{m}^{A}|,|d^{A}_{m+1}|]$, but not both. 2. (2)

If $X$ is a new pre-$a$-clique of size at least three in $(|d_{m}^{A}|,|d^{A}_{m+1}|]$, then every proper subset of $X$ has a new pre-$a$-clique in an interval $(|d_{j}^{A}|,|d^{A}_{j+1}|]$, for some $j<m$.

Lemma 8.2**.**

If $A\subseteq\mathbb{T}_{k}$ has the Strict Witnessing Property and $B\cong A$, then $B$ also has the Strict Witnessing Property.

Proof.

If $B\cong A$ and $A$ has the WP, then $B$ also has the WP by Lemma 5.18. Let $f:A\rightarrow B$ be the strong isomorphism between them. Since $A$ has the SWP, each new pre-clique of size at least two in $A$ is the only new pre-clique occuring in that interval of $A$, hence it is maximal in that interval. By (2) of Definition 8.1, each proper subset of a new pre-clique in a given interval of $A$ occurs as a maximal new pre-clique and is witnessed in some lower interval of $A$. Since $f$ preserves maximal new pre-cliques, each new pre-clique of size at least two in $B$ is a maximal new pre-clique in $B$, and is the only new pre-clique of $B$ in the interval in which it occurs. Thus, $B$ satisfies (2). Furthermore, for any $t\in A$, $f(t)$ is a new singleton pre-$a$-clique in $B$ iff $t$ is a is a new singleton pre-$a$-clique in $A$. Therefore, $B$ has the SWP. ∎

Given a finite tree $A$ with the SWP, we say that $B$ is a copy of $A$ if $A\cong B$. The main theorem of this section, Theorem 8.3, will guarantee a Ramsey Theorem for colorings of copies of a finite tree with the SWP inside a strong coding tree.

Theorem 8.3**.**

Let $T\in\mathcal{T}_{k}$ be a strong coding tree and let $A$ be a finite subtree of $T$ satisfying the Strict Witnessing Property. Then for any coloring of the copies of $A$ in $T$ into finitely many colors, there is a strong coding subtree $S\leq T$ such that all copies of $A$ in $S$ have the same color.

Theorem 8.3 will be proved via four lemmas and an induction argument. The main difficulty is that Case (b) of Theorem 7.2 provides homogeneity for $\operatorname{Ext}_{S}(A,\tilde{X})$ for some strong coding tree $S$; in particular, homogeneity only holds for level sets $X$ end-extending $\tilde{X}$. The issue of new singleton pre-$a$-cliques will be handled similarly to how we handle the case when $\tilde{X}$ has a coding node. We need a strong coding tree in which every $X$ satisfying $A\cup X\cong A\cup\tilde{X}$ has the same color. This will be addressed by the following: Lemma 8.7 will build a fusion sequence to obtain an $S\leq T$ which is homogeneous on $\operatorname{Ext}_{S}(A,Y)$ for each minimal level set $Y$ extending $A_{e}$ such that $A\cup Y\cong A\cup\tilde{X}$. Lemma 8.8 will use a new forcing and arguments from the proof of Theorem 7.2 to obtain a strong coding tree $S\in[B,T]$ in which every $X$ satisfying $A\cup X\cong A\cup\tilde{X}$ has the same color. The last two lemmas involve fusion to construct a strong coding subtree which is homogeneous for the induced color on copies of $A$. The theorem then follows by induction and an application of Ramsey’s Theorem.

The following basic assumption, similar to but stricter than Case (b) of Theorem 7.2, will be used in much of this section.

Assumption 8.4**.**

Let $A$ and $C$ be fixed non-empty finite valid subtrees of a strong coding tree $T\in\mathcal{T}_{k}$ such that

1. (1)

$A$ and $C$ both satisfy the Strict Witnessing Property; and 2. (2)

$C\setminus A$ is a level set containing both a coding node and the sequence $0^{(l_{C})}$.

Let $\tilde{X}$ denote $C\setminus A$, and let $A_{e}$ be the subset of $A^{+}$ which is extended to $\tilde{X}$. Let $d+1$ be the number of nodes in $\tilde{X}$. List the nodes in $A_{e}$ as $\langle s_{i}:i\leq d\rangle$ and the nodes of $\tilde{X}$ as $\langle t_{i}:i\leq d\rangle$ so that each $t_{i}$ extends $s_{i}$ and $t_{d}$ is the coding node in $\tilde{X}$. For $j\in\{0,1\}$, let $I_{j}$ denote the set of $i\leq d$ such that $t_{i}$ has passing number $j$ at $t_{d}$. If $\tilde{X}$ has a new pre-clique over $A$, let $I_{*}$ denote the set of $i\in I$ such that $\{t_{i}:i\in I_{*}\}$ is the new pre-clique in $\tilde{X}$ over $A$. Note that $I_{*}\subseteq I_{1}$ and $t_{d}$ must be among the coding nodes in $C$ witnessing this new pre-cliqe.

For any $X$ such that $A\cup X\cong C$, let $\operatorname{Ext}_{T}(A,X)$ be defined as in equation (45) of Section 7. Thus, $\operatorname{Ext}_{T}(A,X)$ is the collection of level sets $Y\subseteq T$ such that $Y$ end-extends $X$ and $A\cup Y\cong A\cup X$, (equivalently, $A\cup Y\cong C$), and $A\cup Y$ is valid in $T$. Recall that, since $\tilde{X}$ contains a coding node, $A\cup X\cong A\cup\tilde{X}$ implicitly includes that the strong isomorphism from $A\cup\tilde{X}$ to $A\cup X$ preserves passing numbers between $\tilde{X}^{+}$ and $X^{+}$. We hold to the convention that given $Y$ such that $A\cup Y\cong C$, the nodes in $Y$ are labeled $y_{i}$, $i\leq d$, where each $y_{i}\supseteq s_{i}$. In particular, $y_{d}$ is the coding node in $Y$.

In this section, we want to consider all copies of $C$ extending $A$. To that end let

[TABLE]

Now we define the notion of minimal pre-extension, which will be used in the next lemma. For $x\in T$, define $\operatorname{splitpred}_{T}(x)$ to be $x\restriction l$ where $l<|x|$ is maximal such that $x\restriction l$ is a splitting node in $T$.

Definition 8.5** (Minimal pre-extension of $A$ to a copy of $C$).**

Given $A$, $\tilde{X}$, and $C$ as in Assumption 8.4, for $X=\{x_{i}:i\leq d\}$ a level set extending $A_{e}$ such that $x_{i}\supseteq s_{i}$ for each $i\leq d$ and such that $l_{X}$ is the length of some coding node in $T$, we say that $X$ is a minimal pre-extension in $T$ of $A$ to a copy of $C$ if the following hold:

1. (i)

$\{i\leq d:$ the passing number of $x_{i}$ at $x_{d}$ is $1\}=I_{1}$. 2. (ii)

$A\cup\operatorname{SP}_{T}(X)$ satisfies the Strict Witnessing Property, where

[TABLE] 3. (iii)

If $X$ has a new pre-clique over $A$, then $X$ has only one new maximal pre-clique over $A$ which is exactly $\{x_{i}:i\in I_{*}\}\restriction l$, for some $l\in(l_{A},l_{X}]$.

Notice that for (ii) to hold, $\operatorname{SP}_{T}(X)$ must have no new pre-cliques over $A$. Let $\operatorname{MPE}_{T}(A,C)$ denote the set of minimal pre-extenions in $T$ of $A$ to a copy of $C$. When $A$ and $C$ are clear, we call members of $\operatorname{MPE}_{T}(A,C)$ simply minimal pre-extensions. Minimal pre-extensions are exactly the level sets in $T$ which can be extended to a member of $\operatorname{Ext}_{T}(A,\tilde{X})$.

For $X\in\operatorname{MPE}_{T}(A,C)$, define

[TABLE]

Then

[TABLE]

Definition 8.6**.**

A coloring on $\operatorname{Ext}_{T}(A,C)$ is end-homogeneous if for each minimal pre-extension $X$, every member of $\operatorname{Ext}_{T}(A,C;X)$ has the same color.

The following lemma is a slightly modified version of Lemma 6.7 in ****[14]****.

Lemma 8.7** (End-homogeneity).**

Assume 8.4, and let $m$ be the integer such that $\max(A)\subseteq r_{m}(T)$. Then for any coloring $h$ of $\operatorname{Ext}_{T}(A,C)$ into two colors, there is a $T^{\prime}\in[r_{m}(T),T]$ such that $h$ is end-homogeneous on $\operatorname{Ext}_{T^{\prime}}(A,C)$.

Proof.

Let $(m_{j})_{j\in\mathbb{N}}$ enumerate those integers greater than $m$ such that there is a minimal pre-extension of $A$ to a copy of $C$ from among the maximal nodes in $r_{m_{j}}(T)$. Notice that for each $j\in\mathbb{N}$, $\max(r_{m_{j}}(T))$ contains a coding node, although there can be members of $\operatorname{MPE}_{T}(A,C)$ contained in $\max(r_{m_{j}}(T))$ not containing that coding node.

Let $T_{-1}$ denote $T$. Suppose that $j\in\mathbb{N}$ and $T_{j-1}$ is given so that the coloring $h$ is homogeneous on $\operatorname{Ext}_{T_{j-1}}(A,C;X)$ for each minimal pre-extension $X$ in $r_{m_{j}-1}(T_{j-1})$. Let $U_{j-1}$ denote $r_{m_{j}-1}(T_{j-1})$. Enumerate the minimal pre-extensions contained in $\max(r_{m_{j}}(T_{j-1}))$ as $X_{0},\dots,X_{n}$. By induction on $i\leq n$, we will obtain $T_{j}\in[U_{j-1},T_{j-1}]$ such that $\max(r_{m_{j}}(T_{j}))$ end-extends $\max(r_{m_{j}}(T_{j-1}))$ and $\operatorname{Ext}_{T_{j}}(A,C;Z)$ is homogeneous for each minimal pre-extension $Z$ in $\max(r_{m_{j}}(T_{j-1}))$.

Let $l$ denote the length of the nodes in $\max(r_{m_{j}}(T_{j-1}))$, and let $S_{-1}=T_{j-1}$. Suppose $0\leq i\leq n$ and we have strong coding trees $S_{-1},\dots,S_{i-1}$ such that for each $0\leq i^{\prime}\leq i-1$, $S_{i^{\prime}}\in[U_{j-1},S_{i^{\prime}-1}]$ and $h$ is homogeneous on $\operatorname{Ext}_{S_{i^{\prime}}}(A,C;X_{i^{\prime}})$. Note that $X_{i}$ is contained in $r_{m_{j}}(S_{i-1})\restriction l$, though $l$ does not have to be the length of any node in $S_{i-1}$. The point is that the set of nodes $Y_{i}$ in $\max(r_{m_{j}}(S_{i-1}))$ end-extending $X_{i}$ is again a minimal pre-extension. Extend the nodes in $Y_{i}$ to some $Z_{i}\in\operatorname{Ext}_{S_{i-1}}(A,C;Y_{i})$, and let $l^{\prime}$ denote the length of the nodes in $Z_{i}$. Note that $Z_{i}$ has no new pre-cliques over $Y_{i}$. Let $W_{i}$ consist of the nodes in $Z_{i}$ along with the leftmost extensions of the nodes in $\max(r_{m_{j}}(S_{i-1}))\setminus Y_{i}$ to the length $l^{\prime}$ in $S_{i-1}$.

Let $S^{\prime}_{i-1}$ be a strong coding tree in $[U_{j-1},S_{i-1}]$ such that $\max(r_{m_{j}}(S^{\prime}_{i-1}))$ extends $W_{i}$. Such an $S^{\prime}_{i-1}$ exists by Lemmas 6.9 and 6.14 and Theorem 6.16. Apply Case (b) of Theorem 7.2 to obtain a strong coding tree $S_{i}\in[U_{j-1},S^{\prime}_{i-1}]$ such that the coloring on $\operatorname{Ext}_{S_{i}}(A,C;Z_{i})$ is homogeneous. At the end of this process, let $T_{j}=S_{n}$. Note that for each minimal pre-extension $Z\subseteq\max(r_{m_{j}}(T_{j}))$, there is a unique $i\leq n$ such that $Z$ extends $X_{i}$, since each node in $\max(r_{m_{j}}(T_{j}))$ is a unique extension of one node in $\max(r_{m_{j}}(T_{j-1}))$, and hence $\operatorname{Ext}_{T_{j}}(A,C;Z)$ is homogeneous.

Having chosen each $T_{j}$ as above, let $T^{\prime}=\bigcup_{j\in\mathbb{N}}r_{m_{j}}(T_{j})$. Then $T^{\prime}$ is a strong coding tree which is a member of $[r_{m}(T),T]$, and for each minimal pre-extension $Z$ in $T^{\prime}$, $\operatorname{Ext}_{T^{\prime}}(A,C;Z)$ is homogeneous for $h$. Therefore, $h$ is end-homogeneous on $\operatorname{Ext}_{T^{\prime}}(A,C)$. ∎

The next lemma provides a means for uniformizing the end-homogeneity from the previous lemma to obtain one color for all members of $\operatorname{Ext}_{S}(A,C)$. The arguments are often similar to those of Case (a) of Theorem 7.2, but sufficiently different to warrant a proof.

Lemma 8.8**.**

Assume 8.4, and suppose that $B$ is a finite strong coding tree valid in $T$ and $A$ is a subtree of $B$ such that $\max(A)\subseteq\max(B)$. Suppose that $h$ is end-homogeneous on $\operatorname{Ext}_{T}(A,C)$. Then there is an $S\in[B,T]$ such that $h$ is homogeneous on $\operatorname{Ext}_{S}(A,C)$.

Proof.

Given any $U\in[B,T]$, recall that $\operatorname{MPE}_{U}(A,C)$ denotes the set of all minimal pre-extensions of $A$ to a copy of $C$ in $U$. We are under Assumption 8.4. Let $i_{0}\leq d$ be such that $t_{i_{0}}=0^{(l_{C})}$, and note that $i_{0}$ is a member of $I_{0}$. Each member $Y$ of $\operatorname{MPE}_{T}(A,C)$ will be enumerated as $\{y_{i}:i\leq d\}$ so that $y_{i}\supseteq s_{i}$ for each $i\leq d$. Recall notation (68) of $\operatorname{SP}_{T}(Y)$.

Since $C$ satisfies the SWP, $\tilde{X}$ is in $\operatorname{MPE}_{T}(A,C)$. Let $P$ denote $\operatorname{SP}_{T}(\tilde{X})$. Since $\tilde{X}$ is contained in an interval of $T$ above the interval containing $\max(A)$, each node of $P$ extends exactly one node of $A_{e}$. For any $U\in[B,T]$, define $\operatorname{Ext}_{U}(A,P)$ to consist of those $X$ such that

[TABLE]

By assumption, the coloring $h$ on $\operatorname{Ext}_{T}(A,C)$ is end-homogeneous. This induces a coloring $h$ on $\operatorname{MPE}_{T}(A,C)$ by defining, for $Y\in\operatorname{MPE}_{T}(A,C)$, $h(Y)$ to be the $h$-color that all members of $\operatorname{Ext}_{T}(A,C;Y)$ have. This further induces a coloring $h^{\prime}$ on $\operatorname{Ext}_{T}(A,P)$ as follows: For $Q\in\operatorname{Ext}_{T}(A,P)$, for the $Y\in\operatorname{MPE}_{T}(A,C)$ such that $\operatorname{SP}_{T}(Y)=Q$, let $h^{\prime}(Q)=h(Y)$. Given $Q\in\operatorname{Ext}_{T}(A,P)$, the extensions of the $q_{i}\in Q$ such that $i\in I_{1}$ to the level of next coding node in $T$, with passing number $1$ at that coding node, recovers $Y$. Thus, $h^{\prime}$ is well-defined.

Let $L$ denote the collection of all $l\in\mathbb{N}$ such that there is a member of $\operatorname{Ext}_{T}(A,P)$ with maximal nodes of length $l$. For each $i\in(d+1)\setminus\{i_{0}\}$, let $T_{i}=\{t\in T:t\supseteq s_{i}\}$. Let $T_{i_{0}}$ be the collection of all leftmost nodes in $T$ extending $s_{i_{0}}$. Let $\kappa=\beth_{2d+2}$. The following forcing notion $\mathbb{Q}$ will add $\kappa$ many paths through each $T_{i}$, $i\in(d+1)\setminus\{i_{0}\}$ and one path through $T_{i_{0}}$, though with $\kappa$ many labels. The present case is handled similarly to Case (a) of Theorem 7.2.

Let $\mathbb{Q}$ be the set of conditions $p$ such that $p$ is a function of the form

[TABLE]

where $\vec{\delta}_{p}$ is a finite subset of $\kappa$, $l_{p}\in L$, $\{p(i,\delta):\delta\in\vec{\delta}_{p}\}\subseteq T_{i}$ for each $i<d$, and

1. (i)

There is some coding node $c^{T}_{n(p)}$ in $T$ such that $l^{T}_{n(p)}=l_{p}$, and $l^{T}_{n(p)-1}<|p(i,\delta)|\leq l_{p}$ for each $(i,\delta)\in(d+1)\times\vec{\delta}_{p}$. 2. (ii)

1. ($\alpha$)

If $i\in I_{1}$, then $p(i,\delta)=\operatorname{splitpred}_{T}(y)$ for some $y\in T_{i}\restriction l_{p}$. 2. $(\beta)$

If $i\in I_{0}$, then $p(i,\delta)\in T_{i}\restriction l_{p}$ and has immediate extension [math] in $T$.

It follows from the definition that for $p\in\mathbb{Q}$, $\operatorname{ran}(p):=\{p(i,\delta):(i,\delta)\in(d+1)\times\vec{\delta}_{p}\}$ is free in $T$: leftmost extensions add no new pre-cliques. Furthermore, all nodes in $\operatorname{ran}(p)$ are contained in the $n(p)$-th interval of $T$. We point out that $\operatorname{ran}(p)$ may or may not contain a coding node. If it does, then that coding node must appear as $p(i,\delta)$ for some $i\in I_{0}$; this $i$ may or may not equal $d$.

The partial ordering on $\mathbb{Q}$ is defined as follows: $q\leq p$ if and only if $l_{q}\geq l_{p}$, $\vec{\delta}_{q}\supseteq\vec{\delta}_{p}$,

1. (i)

$q(i,\delta)\supseteq p(i,\delta)$ for each $(i,\delta)\in(d+1)\times\vec{\delta}_{p}$; and 2. (ii)

$\operatorname{ran}(q\restriction\vec{\delta}_{p}):=\{q(i,\delta):(i,\delta)\in(d+1)\times\vec{\delta}_{p}\}$ has no new pre-cliques over $\operatorname{ran}(p)$.

By arguments similar to those in the proof of Theorem 7.2, $(\mathbb{Q},\leq)$ is an atomless partial order, and any condition in $\mathbb{Q}$ can be extended by two incompatible conditions of length greater than any given $l\in\mathbb{N}$.

Let $\dot{\mathcal{U}}$ be a $\mathbb{Q}$-name for a non-principal ultrafilter on $L$. For each $i\leq d$ and $\alpha<\kappa$, let $\dot{b}_{i,\alpha}=\{\langle p(i,\alpha),p\rangle:p\in\mathbb{Q}$ and $\alpha\in\vec{\delta}_{p}\}$, a $\mathbb{Q}$-name for the $\alpha$-th generic branch through $T_{i}$. For any condition $p\in\mathbb{Q}$, for $(i,\alpha)\in I_{0}\times\vec{\delta}_{p}$, $p$ forces that $\dot{b}_{i,\alpha}\restriction l_{p}=p(i,\alpha)$. For $(i,\alpha)\in I_{1}\times\vec{\delta}_{p}$, $p$ forces that $\operatorname{splitpred}_{T}(\dot{b}_{i,\alpha}\restriction l_{p})=p(i,\alpha)$. For $\vec{\alpha}=\langle\alpha_{0},\dots,\alpha_{d}\rangle\in[\kappa]^{d+1}$,

[TABLE]

For $l\in L$, we shall use the abbreviation

[TABLE]

which is exactly $\{\dot{b}_{i,\alpha_{i}}\restriction l:i\in I_{0}\}\cup\{\operatorname{splitpred}_{T}(\dot{b}_{i,\alpha_{i}}\restriction l):i\in I_{1}\}$.

Similarly to the proof of Theorem 7.2, we will find infinite pairwise disjoint sets $K_{i}\subseteq\kappa$, $i\leq d$, such that $K_{0}<K_{1}<\dots<K_{d}$, and conditions $p_{\vec{\alpha}}$, $\vec{\alpha}\in\prod_{i\leq d}K_{i}$, such that these conditions are pairwise compatible, have the same images in $T$, and force the same color $\varepsilon^{*}$ for $h^{\prime}(\dot{b}_{\vec{\alpha}}\restriction l)$ for $\dot{\mathcal{U}}$ many levels $l$ in $L$. Moreover, the nodes $\{t^{*}_{i}:i\leq d\}$ obtained from the application of the Erdős-Rado Theorem for this setting will extend $\{s_{i}:i\leq d\}$ and form a member of $\operatorname{Ext}_{T}(A,P)$. The arguments are quite similar to those in Theorem 7.2, so we only fill in the details for arguments which are necessarily different.

Part I**.** Given $p\in\mathbb{Q}$ and $\vec{\alpha}\in[\vec{\delta}_{p}]^{d+1}$, let

[TABLE]

For each $\vec{\alpha}\in[\kappa]^{d+1}$, choose a condition $p_{\vec{\alpha}}\in\mathbb{Q}$ such that

1. (1)

$\vec{\alpha}\subseteq\vec{\delta}_{p_{\vec{\alpha}}}$. 2. (2)

$P(p,\vec{\alpha})\in\operatorname{Ext}_{T}(A,P)$. 3. (3)

There is an $\varepsilon_{\vec{\alpha}}\in 2$ such that $p_{\vec{\alpha}}\Vdash$ “$h(\dot{b}_{\vec{\alpha}}\restriction l)=\varepsilon_{\vec{\alpha}}$ for $\dot{\mathcal{U}}$ many $l$ in $L$.” 4. (4)

$h^{\prime}(P(p_{\vec{\alpha}},\vec{\alpha}))=\varepsilon_{\vec{\alpha}}$.

Properties (1) - (4) can be guaranteed as follows. For each $i\leq d$, let $u_{i}$ denote the member of $P$ which extends $s_{i}$. For each $\vec{\alpha}\in[\kappa]^{d+1}$, let

[TABLE]

Then $p^{0}_{\vec{\alpha}}$ is a condition in $\mathbb{P}$ and $\vec{\delta}_{p_{\vec{\alpha}}^{0}}=\vec{\alpha}$, so (1) holds for every $p\leq p^{0}_{\vec{\alpha}}$. Further, $\operatorname{ran}(p^{0}_{\vec{\alpha}})$ is a member of $\operatorname{Ext}_{T}(A,P)$ since it equals $P$. For any $p\leq p_{\vec{\alpha}}^{0}$, (ii) of the definition of the partial ordering on $\mathbb{Q}$ guarantees that $P(p,\vec{\alpha})$ has no new pre-cliques over $\operatorname{ran}(p)$, and hence is also a member of $\operatorname{Ext}_{T}(A,P)$. Thus, (2) holds for any $p\leq p_{\vec{\alpha}}^{0}$. Take an extension $p^{1}_{\vec{\alpha}}\leq p^{0}_{\vec{\alpha}}$ which forces $h^{\prime}(\dot{b}_{\vec{\alpha}}\restriction l)$ to be the same value for $\dot{\mathcal{U}}$ many $l\in L$, and which decides that value, denoted by $\varepsilon_{\vec{\alpha}}$. Then any $p\leq p^{1}_{\vec{\alpha}}$ satisfies (3).

Take $p_{\vec{\alpha}}^{3}\leq p^{2}_{\vec{\alpha}}$ which decides $h^{\prime}(\dot{b}_{\vec{\alpha}}^{3}\restriction l)=\varepsilon_{\vec{\alpha}}$, for some $l$ such that $l_{p^{2}_{\vec{\alpha}}}<l\leq l_{p^{3}_{\vec{\alpha}}}$. If $l=l_{p^{3}_{\vec{\alpha}}}$, let $p_{\vec{\alpha}}=p_{\vec{\alpha}}^{3}$. Otherwise, let $\vec{\delta}_{\vec{\alpha}}=\vec{\delta}_{p^{2}_{\vec{\alpha}}}$ and define $p_{\vec{\alpha}}$ as follows: For each $i\in I_{0}$, for $\delta\in\vec{\delta}_{\vec{\alpha}}$, let $p_{\vec{\alpha}}(i,\delta)=p_{\vec{\alpha}}^{3}(i,\delta)\restriction l$. For each $i\in I_{1}$, for $\delta\in\vec{\delta}_{\vec{\alpha}}$, let $p_{\vec{\alpha}}(i,\delta)=\operatorname{splitpred}_{T}(p_{\vec{\alpha}}^{3}(i,\delta)\restriction l)$. Then $p_{\vec{\alpha}}$ is a condition in $\mathbb{Q}$, and $p_{\vec{\alpha}}\leq p_{\vec{\alpha}}^{2}$, so it satisfies (1) - (3). Furthermore, $h^{\prime}(P(p_{\vec{\alpha}},\vec{\alpha}))=\varepsilon_{\vec{\alpha}}$, so $p_{\vec{\alpha}}$ satisfies (4).

We are assuming $\kappa=\beth_{2d+2}$. Let $D_{e}=\{0,2,\dots,2d\}$ and $D_{o}=\{1,3,\dots,2d+1\}$, the sets of even and odd integers less than $2d+2$, respectively. Let $\mathcal{I}$ denote the collection of all functions $\iota:(2d+2)\rightarrow(2d+2)$ such that $\iota\restriction D_{e}$ and $\iota\restriction D_{o}$ are strictly increasing sequences and $\{\iota(0),\iota(1)\}<\{\iota(2),\iota(3)\}<\dots<\{\iota(2d),\iota(2d+1)\}$. For $\vec{\theta}\in[\kappa]^{2d+2}$, $\iota(\vec{\theta}\,)$ determines the pair of sequences of ordinals $(\theta_{\iota(0)},\theta_{\iota(2)},\dots,\theta_{\iota(2d))}),(\theta_{\iota(1)},\theta_{\iota(3)},\dots,\theta_{\iota(2d+1)})$, both of which are members of $[\kappa]^{d+1}$. Denote these as $\iota_{e}(\vec{\theta}\,)$ and $\iota_{o}(\vec{\theta}\,)$, respectively. Let $\vec{\delta}_{\vec{\alpha}}$ denote $\vec{\delta}_{p_{\vec{\alpha}}}$, $k_{\vec{\alpha}}$ denote $|\vec{\delta}_{\vec{\alpha}}|$, and let $l_{\vec{\alpha}}$ denote $l_{p_{\vec{\alpha}}}$. Let $\langle\delta_{\vec{\alpha}}(j):j<k_{\vec{\alpha}}\rangle$ denote the enumeration of $\vec{\delta}_{\vec{\alpha}}$ in increasing order. Define a coloring $f$ on $[\kappa]^{2d+2}$ into countably many colors as follows: Given $\vec{\theta}\in[\kappa]^{2d+2}$ and $\iota\in\mathcal{I}$, to reduce the number of subscripts, letting $\vec{\alpha}$ denote $\iota_{e}(\vec{\theta}\,)$ and $\vec{\beta}$ denote $\iota_{o}(\vec{\theta}\,)$, define

[TABLE]

Let $f(\vec{\theta}\,)$ be the sequence $\langle f(\iota,\vec{\theta}\,):\iota\in\mathcal{I}\rangle$, where $\mathcal{I}$ is given some fixed ordering. By the Erdős-Rado Theorem, there is a subset $K\subseteq\kappa$ of cardinality $\aleph_{1}$ which is homogeneous for $f$.

Take $K^{\prime}\subseteq K$ such that between each two members of $K^{\prime}$ there is a member of $K$. Then take subsets $K_{i}\subseteq K^{\prime}$ such that $K_{0}<\dots<K_{d}$ and each $|K_{i}|=\aleph_{0}$. The following four lemmas are direct analogues of Lemmas 7.4, 7.5, 7.6, and 7.7. Their proofs follow by simply making the correct notational substitutions, and so are omitted.

Lemma 8.9**.**

There are $\varepsilon^{*}\in 2$, $k^{*}\in\omega$, and $\langle t_{i,j}:j<k^{*}\rangle$, $i\leq d$, such that for all $\vec{\alpha}\in\prod_{i\leq d}K_{i}$ and each $i\leq d$, $\varepsilon_{\vec{\alpha}}=\varepsilon^{*}$, $k_{\vec{\alpha}}=k^{*}$, and $\langle p_{\vec{\alpha}}(i,\delta_{\vec{\alpha}}(j)):j<k_{\vec{\alpha}}\rangle=\langle t_{i,j}:j<k^{*}\rangle$.

Let $l^{*}=|t_{i_{0}}|$. Then for each $i\in I_{0}$, the nodes $t_{i,j}$, $j<k^{*}$, have length $l^{*}$; and for each $i\in I_{1}$, the nodes $t_{i,j}$, $j<k^{*}$, have length in the interval $(l^{T}_{n-1},l^{T}_{n})$, where $n$ is the index of the coding node in $T$ of length $l^{*}$.

Lemma 8.10**.**

Given any $\vec{\alpha},\vec{\beta}\in\prod_{i\leq d}K_{i}$, if $j,k<k^{*}$ and $\delta_{\vec{\alpha}}(j)=\delta_{\vec{\beta}}(k)$, then $j=k$.

For any $\vec{\alpha}\in\prod_{i\leq d}K_{i}$ and any $\iota\in\mathcal{I}$, there is a $\vec{\theta}\in[K]^{2d+2}$ such that $\vec{\alpha}=\iota_{o}(\vec{\theta})$. By homogeneity of $f$, there is a strictly increasing sequence $\langle j_{i}:i\leq d\rangle$ of members of $k^{*}$ such that for each $\vec{\alpha}\in\prod_{i\leq d}K_{i}$, $\delta_{\vec{\alpha}}(j_{i})=\alpha_{i}$. For each $i\leq d$, let $t^{*}_{i}$ denote $t_{i,j_{i}}$. Then for each $i\leq d$ and each $\vec{\alpha}\in\prod_{i\leq d}K_{i}$,

[TABLE]

Lemma 8.11**.**

For any finite subset $\vec{J}\subseteq\prod_{i\leq d}K_{i}$, the set of conditions $\{p_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\,\}$ is compatible. Moreover, $p_{\vec{J}}:=\bigcup\{p_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\,\}$ is a member of $\mathbb{P}$ which is below each $p_{\vec{\alpha}}$, $\vec{\alpha}\in\vec{J}$.

Lemma 8.12**.**

If $\beta\in\bigcup_{i\leq d}K_{i}$, $\vec{\alpha}\in\prod_{i\leq d}K_{i}$, and $\beta\not\in\vec{\alpha}$, then $\beta$ is not a member of $\vec{\delta}_{\vec{\alpha}}$.

Part II**.** Let $(n_{j})_{j\in\mathbb{N}}$ denote the set of indices for which there is an $X\in\operatorname{MPE}_{T}(A,C)$ with $X=\max(V)$ for some $V$ of $r_{n_{j}}[B,T]$. For $i\in I_{0}$, let $u^{*}_{i}=t^{*}_{i}$. For $i\in I_{1}$, let $u_{i}^{*}$ be the leftmost extension of $t^{*}_{i}$ in $T\restriction l^{*}$. Note that $\{u_{i}^{*}:i\leq d\}$ has no new pre-cliques over $A_{e}$, since leftmost extensions of splitting nodes with no new pre-cliques add no new pre-cliques; this follows from the WP of $T$. Extend each node $u$ in $B^{+}\setminus A_{e}$ to its leftmost extension in $T\restriction l^{*}$ and label that extension $u^{*}$. Let

[TABLE]

Then $U^{*}$ extends $B^{+}$, and $U^{*}$ has no new pre-cliques over $B$. Let $n_{-1}$ be the integer such that $B=r_{n_{-1}}(B)$. Take $S_{0}\in r_{n_{0}}[B,T]$ such that the nodes in $\max(r_{n_{-1}+1}(S_{0}))$ extend the nodes in $U^{*}$. This is possible by Lemma 6.18.

Suppose that $j\in\mathbb{N}$, and for all $i<j$, we have chosen $S_{i}\in r_{n_{i}}[B,T]$ such that $i<i^{\prime}<j$ implies $S_{i}\sqsubset S_{i^{\prime}}$, and $h^{\prime}$ is constant of value $\varepsilon^{*}$ on $\operatorname{Ext}_{S_{i}}(A,P)$. Take $V_{j}\in r_{n_{j}}[S_{j-1},T]$, and let $X$ denote $\max(V_{j})$. Notice that each member of $\operatorname{Ext}_{X}(A,P)$ extends the nodes in $U^{*}$. By the definition of $n_{j}$, the set of nodes $X$ contains a coding node. For each $i\in I_{0}$, let $Y_{i}$ denote the set of all $t\in T_{i}\cap X$ which have immediate extension [math] in $T$. Let $n$ be such that $l_{X}=l^{T}_{n}$. For each $i\in I_{1}$, let $Y_{i}$ denote the set of all splitting predecessors of nodes in $T_{i}\cap X$ which split in the interval $(l^{T}_{n-1},l^{T}_{n}]$ of $T$. For each $i\leq d$, let $J_{i}$ be a subset of $K_{i}$ of size $|Y_{i}|$, and enumerate the members of $Y_{i}$ as $q(i,\delta)$, $\delta\in J_{i}$. Let $\vec{J}$ denote the set of $\vec{\alpha}\in\prod_{i\leq d}J_{i}$ such that the set $\{q(i,\alpha_{i}):i\leq d\}$ has no new pre-cliques over $A$. Thus, the collection of sets $\{q(i,\alpha_{i}):i\leq d\}$, $\vec{\alpha}\in\vec{J}$, is exactly the collection of sets of nodes in the interval $(l^{T}_{n-1},l^{T}_{n}]$ of $T$ which are members of $\operatorname{Ext}_{T}(A,P)$. Moreover, for $\vec{\alpha}\in\vec{J}$ and $i\leq d$,

[TABLE]

To complete the construction of the desired $q\in\mathbb{Q}$ for which $q\leq p_{\vec{\alpha}}$ for all $\vec{\alpha}\in\vec{J}$, let $\vec{\delta}_{q}=\bigcup\{\vec{\delta}_{\vec{\alpha}}:\vec{\alpha}\in\vec{J}\}$. For each pair $(i,\gamma)$ with $\gamma\in\vec{\delta}_{q}\setminus J_{i}$, there is at least one $\vec{\alpha}\in\vec{J}$ and some $m<k^{*}$ such that $\gamma=\delta_{\vec{\alpha}}(m)$. As in Case (a) of Theorem 7.2, for any other $\vec{\beta}\in\vec{J}$ for which $\gamma\in\vec{\delta}_{\vec{\beta}}$, it follows that $p_{\vec{\beta}}(i,\gamma)=p_{\vec{\alpha}}(i,\gamma)=t^{*}_{i,m}$ and $\delta_{\vec{\beta}}(m)=\gamma$. If $i\in I_{0}$, let $q(i,\gamma)$ be the leftmost extension of $t_{i,m}^{*}$ in $T\restriction l^{V_{j}}_{n_{j}}$. If $i\in I_{1}$, let $q(i,\gamma)$ be the leftmost extension of $t^{*}_{i,m}$ to a splitting node in $T$ in the interval $(l^{V_{j}}_{n_{j}-1},l^{V_{j}}_{n_{j}}]$. Such a splitting node must exist, because the coding node in $X$ must have no pre-cliques with $t^{*}_{i}$ (since $i\in I_{1}$). Thus, by Lemma 6.9 the leftmost extension of $t^{*}_{i}$ in $T$ to length $l_{X}$ has no pre-cliques with the coding node in $X$, so it has a splitting predecessor in the interval $(l^{V_{j}}_{n_{j}-1},l^{V_{j}}_{n_{j}}]$. Define

[TABLE]

By a proof similar to that of Claim 3, it follows that $q\leq p_{\vec{\alpha}}$, for each $\vec{\alpha}\in\vec{J}$.

Take an $r\leq q$ in $\mathbb{P}$ which decides some $l_{j}$ in $L$, and such that for all $\vec{\alpha}\in\vec{J}$, $h^{\prime}(\dot{b}_{\vec{\alpha}}\restriction l_{j})=\varepsilon^{*}$. Without loss of generality, we may assume that the maximal nodes in $r$ have length $l_{j}$. If $q(i^{\prime},\alpha^{\prime})$ is a coding node for some $i^{\prime}\in I_{0}$ and $\alpha^{\prime}\in J_{i^{\prime}}$, then let $c_{r}$ denote $r(i^{\prime},\alpha^{\prime})$; otherwise, let $c_{r}$ denote the leftmost extension in $T$ of the coding node in $X$ to length $l_{j}$. Let $c_{X}$ denote the coding node in $X$.

Let $Z_{0}$ denote those nodes in $\operatorname{SP}_{T}(X)$ which have length equal to $l_{X}$ and are not in $\bigcup_{i\in I_{0}}Y_{i}$. For each $z\in Z_{0}$, let $s_{z}$ denote the leftmost extension of $z$ in $T$ to length $l_{j}$. Let $Z_{1}$ denote the set of all nodes in $\operatorname{splitpred}_{T}(X)$ which are not in $Z_{0}\cup\bigcup_{i\in I}Y_{i}$. For each $z\in Z_{1}$, let $s_{z}$ denote the splitting predecessor of the leftmost extension of $z$ in $T$ to length $l_{j}$. This splitting node $s_{z}$ exists in $T$ for the following reason: If $z$ is a splitting predecessor of a node in $X$, then $z$ has no pre-$k$-clique with $c_{X}$, so the leftmost extension of $z$ to any length has no pre-$k$-cliques with any extension of $c_{X}$. In particular, the set $\{s_{z}:z\in Z_{0}\cup Z_{1}\}$ has no new pre-cliques over $X$.

Let

[TABLE]

Let $Z^{*}$ denote all extensions in $T$ of the members of $Z^{-}$ to length $l_{j}$. Note that $Z^{-}=\operatorname{splitpred}_{T}(Z^{*})$, which end-extends $\operatorname{splitpred}_{T}(X)$. Let $m$ denote the index such that the maximal coding node in $V_{j}$ below $c_{X}$ is $c^{V_{j}}_{n_{m}}$. Note that $Z^{*}$ has no new pre-cliques over $\operatorname{splitpred}_{T}(X)$; furthermore, the tree induced by $r_{n_{m}}(V_{j})\cup Z^{*}$ is strongly similar to $V_{j}$, except that the coding node might possibly be in the wrong place. Using Lemma 6.18, there is an $S_{j}\in r_{n_{j}}[r_{n_{m}}(V_{j}),T]$ with $\max(S_{j})$ extending $Z^{*}$. Then every member of $\operatorname{Ext}_{S_{j}}(A,P)$ has the same $h^{\prime}$ color $\varepsilon^{*}$, by the choice of $r$, since each minimal pre-extension in $\operatorname{MPE}_{S_{j}}(A,C)$ extends some member of $\operatorname{Ext}_{S_{j}}(A,P)$ which extends members in $\operatorname{ran}(r)$ and so have $h^{\prime}$-color $\varepsilon^{*}$.

Let $S=\bigcup_{j\in\mathbb{N}}S_{j}$. Then $S$ is a strong coding tree in $[B,T]$. Given any $Y\in\operatorname{Ext}_{S}(A,C)$, there is some $X\in\operatorname{MPE}_{S}(A,C)$ such that $Y$ extends $X$. Since $\operatorname{SP}_{S}(X)$ is in $\operatorname{Ext}_{S_{j}}(A,P)$ for some $j\in\mathbb{N}$, $\operatorname{SP}_{S}(X)$ has $h^{\prime}$ color $\varepsilon^{*}$. Thus, $Y$ has $h$-color $\varepsilon^{*}$. ∎

Lemma 8.13**.**

Assume 8.4. Then there is a strong coding subtree $S\leq T$ such that for each copy $A^{\prime}$ of $A$ in $S$, $h$ is homogeneous on $\operatorname{Ext}_{S}(A^{\prime},C)$.

Proof.

Let $(n_{i})_{i\in\mathbb{N}}$ be the sequence of integers such that $r_{n_{i}}(T)$ contains a copy of $A$ which is valid in $r_{n_{i}}(T)$ and such that $\max(A)\subseteq\max(r_{n_{i}}(T))$. Let $n_{-1}=0$, $T_{-1}=T$, and $U_{-1}=r_{0}(T)$.

Suppose $i\in\mathbb{N}$, and $U_{i-1}\cong r_{n_{i-1}}(T)$ and $T_{i-1}$ are given satisfying that for each copy $A^{\prime}$ of $A$ valid in $U_{i-1}$ with $\max(A)\subseteq\max(U_{i-1})$, $h$ is homogeneous on $\operatorname{Ext}_{U_{i-1}}(A^{\prime},C)$. Let $U_{i}$ be in $r_{n_{i}}[U_{i-1},T_{i-1}]$. Enumerate all copies $A^{\prime}$ of $A$ which are valid in $U_{i}$ and have $\max(A^{\prime})\subseteq\max(U_{i})$ as $\langle A_{0},\dots,A_{m}\rangle$. Apply Lemma 8.7 to obtain $R_{0}\in[U_{i},T_{i-1}]$ which is end-homogeneous for $\operatorname{Ext}_{R_{0}}(A_{0},C)$. Then apply Lemma 8.8 to obtain $R^{\prime}_{0}\in[U_{i},R_{0}]$ such that $\operatorname{Ext}_{R^{\prime}_{0}}(A_{0},C)$ is homogeneous for $h$. Given $R^{\prime}_{j}$ for $j<m$, apply Lemma 8.7 to obtain a $R_{j+1}\in[U_{i},R^{\prime}_{j}]$ which is end-homogeneous for $\operatorname{Ext}_{R_{j+1}}(A_{j+1},C)$. Then apply Lemma 8.8 to obtain $R^{\prime}_{j+1}\in[U_{i},R_{j+1}]$ such that $\operatorname{Ext}_{R^{\prime}_{j+1}}(A_{j+1},C)$ is homogeneous for $h$. Let $T_{i}=R^{\prime}_{m}$.

Let $U=\bigcup_{i\in\mathbb{N}}U_{i}$. Then $U\leq T$ and $h$ has the same color on $\operatorname{Ext}_{U}(A^{\prime},C)$ for each copy $A^{\prime}$ of $A$ which is valid in $U$. Finally, take $S\leq U$ such that for each $m\in\mathbb{N}$, $r_{m}(S)$ is valid in $U$. Then each copy $A^{\prime}$ of $A$ in $S$ is valid in $U$. Hence, $h$ is homogeneous on $\operatorname{Ext}_{S}(A^{\prime},C)$, for each copy $A^{\prime}$ of $A$ in $S$. ∎

For the setting of Case (a) in Theorem 7.2, a similar lemma holds. The proof is omitted, as it is almost identical, making the obvious changes.

Lemma 8.14**.**

Let $T$ be a member of $\mathcal{T}_{k}$, let $A,\tilde{X},h$ be as in Case (a) of Theorem 7.2, and let $C=A\cup\tilde{X}$. Then there is a strong coding tree $S\leq T$ such that for each $A^{\prime}\subseteq S$ with $A^{\prime}\cong A$, $\operatorname{Ext}_{S}(A^{\prime},C)$ is homogeneous for $h$.

Finally, for the case of $k\geq 4$, a new phenomenon appears: new singleton pre-$a$-cliques for $a\geq 4$ must be dealt with. The steps are very similar to the case when the level set $\tilde{X}$ contains a coding node.

Case (b*′***). **Assume $k\geq 4$. Let $A$ and $C$ be fixed non-empty finite valid subtrees of a strong coding tree $T\in\mathcal{T}_{k}$ such that

1. (1)

$A$ and $C$ both satisfy the Strict Witnessing Property; and 2. (2)

$C\setminus A$ is a level set containing exactly one new singleton pre-$a$-clique, for some $a\in[4,k]$.

Given a copy $A^{\prime}$ of $A$ in a strong $\mathcal{H}_{k}$-coding tree $T$, let $\operatorname{Ext}_{T}(A^{\prime},C)$ denote the set of all $C^{\prime}$ contained in $T$ which end-extend $A^{\prime}$ and such that $C^{\prime}\cong A^{\prime}$.

Lemma 8.15** (Case (b′)).

Given $A$ and $C$ as above, there is a strong coding subtree $S\leq T$ such that for each copy $A^{\prime}$ of $A$ in $S$, $h$ is homogeneous on $\operatorname{Ext}_{S}(A^{\prime},C)$.

Proof.

The proof follows from simple modifications of the proofs of Case (b) in Theorem 7.2 and Lemmas 8.7, 8.8, and 8.13. Just replace the coding node in Case (b) with the new singleton pre-$a$-clique in Case (b*′*). Splitting predecessors work as before. ∎

Proof of Theorem 8.3. The proof is by induction on the number of critical nodes, where in this proof, by critical node we mean a coding node, a splitting node, or a new singleton pre-$a$-clique for some $a\in[4,k]$. Suppose first that $A$ consists of a single node. Then $A$ consists of a single splitting node in $T$ on the leftmost branch of $T$, so the strongly isomorphic copies of $A$ are exactly the leftmost splitting nodes in $T$. Recall that $\mathcal{S}[0]$ denotes the set of all finite length sequences of [math]’s. Thus, the copies of $A$ in $T$ are exactly those splitting nodes in $T$ which are members of $\mathcal{S}[0]$. Let $h$ be any finite coloring on the splitting nodes in the leftmost branch of $T$. By Ramsey’s Theorem, infinitely many splitting nodes in the leftmost branch of $T$ must have the same $h$ color. By the Extension Lemmas in Section 6, there is a subtree $S\leq T$ in which all splitting nodes in the leftmost branch of $S$ have the same $h$ color.

Now assume that $n\geq 1$ and the theorem holds for each finite tree $B$ with $n$ or less critical nodes such that $B$ satisfies the SWP and $\max(B)$ contains a node which is a sequence of all [math]’s. Let $C$ be a finite tree with $n+1$ critical nodes containing a maximal node in $\mathcal{S}[0]$, and suppose $h$ maps the copies of $C$ in $T$ into finitely many colors. Let $d$ denote the maximal critical node in $C$ and let $B=\{t\in C:|t|<|d|\}$. Apply Lemma 8.13, 8.14 or 8.15, as appropriate, to obtain $T^{\prime}\leq T$ so that for each copy $V$ of $B$ in $T^{\prime}$, the set $\operatorname{Ext}_{T^{\prime}}(V,C)$ is homogeneous for $h$. Define $g$ on the copies of $B$ in $T^{\prime}$ by letting $g(V)$ be the value of $h$ on $V\cup X$ for any $X\in\operatorname{Ext}_{T^{\prime}}(V,C)$. By the induction hypothesis, there is an $S\leq T^{\prime}$ such that $g$ is homogeneous on all copies of $B$ in $S$. It follows that $h$ is homogeneous on the copies of $C$ in $S$.

To finish, let $A$ be any finite tree satisfying the SWP. If $\max(A)$ does not contain a member of $\mathcal{S}[0]$, let $l_{A}$ denote the longest length of nodes in $A$, and let $\tilde{A}$ be the tree induced by $A\cup\{0^{(l_{A})}\}$. Otherwise, let $\tilde{A}=A$. Let $g$ be a finite coloring of the copies of $A$ in $T$. To each copy $B$ of $\tilde{A}$ in $T$ there corresponds a unique copy of $A$ in $T$, denoted $\varphi(B)$: If $\tilde{A}=A$, then $\varphi(B)=B$; if $\tilde{A}\neq A$, then $\varphi(B)$ is $B$ with the leftmost node in $\max(B)$ removed. For each copy $B$ of $\tilde{A}$, define $h(B)=g(\varphi(B))$. Take $S\leq T$ homogeneous for $h$. Then $S$ is homogeneous for $g$ on the copies of $A$ in $S$. $\square$

9. Main Ramsey Theorem for strong $\mathcal{H}_{k}$-coding trees

The third phase of this article takes place in this and the next section. Subsection 9.1 develops the notion of incremental trees, which sets the stage for envelopes for incremental antichains. These envelopes transform finite antichains of coding nodes to finite trees with the Strict Witnessing Property, enabling applications of Theorem 8.3 to deduce Theorem 9.9. This theorem takes a finite coloring of all antichains of coding nodes strictly similar to a given finite antichain of coding nodes and finds a strong coding tree in which the coloring has one color. After showing in Lemma 10.1 that any strong coding tree contains an antichain of coding nodes coding a Henson graph, we will apply Theorem 9.9 to prove that each Henson graph has finite big Ramsey degrees, thus obtaining the main result of this paper in Theorem 10.2.

9.1. Incremental trees

The new notions of incremental new pre-cliques and incrementally witnessed pre-cliques, and incremental trees are defined now. The main lemma of this subsection, Lemma 9.3, shows that given a strong coding tree $T$, there is an incremental strong coding subtree $S\leq T$ and a set $W\subseteq T$ of coding nodes disjoint from $S$ such that all pre-cliques in $S$ are incrementally witnessed by coding nodes in $W$. This sets the stage for the development of envelopes with the Strict Witnessing Property in the next subsection.

Definition 9.1** (Incremental Pre-Cliques).**

Let $S$ be a subtree of $\mathbb{T}_{k}$, and let $\langle l_{j}:j<\tilde{j}\rangle$ list in increasing order the minimal lengths of new pre-cliques in $S$, except for singleton new pre-$3$-cliques, where $\tilde{j}\in\mathbb{N}$ or $\tilde{j}=\mathbb{N}$. We say that $S$ has incremental new pre-cliques, or simply $S$ is incremental, if letting

[TABLE]

the following hold for each $j<\tilde{j}$:

1. (1)

$S_{l_{j},1}$ is a new pre-$a$-clique for some $a\in[3,k]$, and no proper subset of $S_{l_{j},1}$ is a new pre-$b$-clique for any $b\in[3,k]$; 2. (2)

If $a=3$ and $S_{l_{j},1}$ has more than two members, then for each proper subset $X\subsetneq S_{l_{j},1}$ of size at least $2$, for some $i<j$, $X\restriction l_{i}=S_{l_{i},1}$ and is also a pre-$3$-clique; 3. (3)

If $a>3$ and $S_{l_{j},1}$ has at least two members, then for each proper subset $X\subsetneq S_{l_{j},1}$, for some $i<j$, $X\restriction l_{i}=S_{l_{i},1}$ and is also a pre-$a$-clique; 4. (4)

If $a>3$, then there are $l_{j-1}<l^{3}<\dots<l^{a}=l_{j}$ such that for each $3\leq b\leq a$, $S_{l_{j},1}\restriction l^{b}$ is a pre-$b$-clique. Furthermore, for some $m$, $|d^{S}_{m}|<l^{3}<l^{a}=l_{j}<|d^{S}_{m+1}|$.

A tree $S\in\mathcal{T}_{k}$ is called an incremental strong coding tree if $S$ is incremental and moreover, the node $d^{S}_{m+1}$ in (4) is a coding node in $S$.

Note that every subtree of an incremental strong coding tree is incremental, but a strong coding subtree of an incremental strong coding tree need not be an incremental strong coding tree. Note also that in (4), the pre-$b$-cliques for $3\leq b<a$ at the levels $l^{3}$ through $l^{a}$ are not new, but they build up to the new pre-$a$-clique in the interval $(l_{j-1},l_{j}]$. This redundancy will actually make the definition of the envelopes simpler.

Definition 9.2** (Incrementally Witnessed Pre-Cliques).**

Let $S,T\in\mathcal{T}_{k}$ be such that $S$ is incremental and $S\leq T$. We say that the pre-cliques in $S$ are incrementally witnessed by a set of witnessing coding nodes $W\subseteq T$ if the following hold. Given that $\langle l_{j}:j\in\mathbb{N}\rangle$ is the increasing enumeration of the minimal lengths of new pre-cliques in $S$, for each $j\in\mathbb{N}$ the following hold:

1. (1)

$|d^{S}_{m_{n}-1}|<l_{j}<l^{S}_{n}$ for some $n\in\mathbb{N}$. 2. (2)

If $S_{l_{j},1}$ is a new pre-$a_{j}$-clique of size at least two, where $a_{j}\in[3,k]$, then there exist coding nodes $w_{j}^{3},\dots,w_{j}^{a_{j}}$ in $T$ such that, letting $W$ denote $\bigcup_{j\in\mathbb{N}}\{w_{j}^{3},\dots w_{j}^{a_{j}}\}$, the set of all these witnessing coding nodes,

1. (a)

The set of nodes $\{w_{j}^{3},\dots,w_{j}^{a_{j}}:j\in\mathbb{N}\}$ forms a pre-$(a_{j}-2)$-clique which witnesses the pre-$a_{j}$-clique in $S_{l_{j},1}$. 2. (b)

The nodes in $\{w_{j}^{3},\dots,w_{j}^{a_{j}}\}$ do not form pre-cliques with any nodes in $(W\setminus\{w_{j}^{3},\dots,w_{j}^{a_{j}}\})\cup(S\restriction|w^{a_{j}}_{j}|\setminus S_{l_{j},1}^{\prime})$ where $S_{l_{j},1}^{\prime}$ denotes the set of nodes in $S\restriction|w^{a_{j}}_{j}|$ which end-extend $S_{l_{j},1}$. 3. (c)

If $Z\subseteq\{w_{j}^{3},\dots,w_{j}^{a_{j}}\}\cup(S\restriction|w^{a_{j}}_{j}|)$ forms a pre-clique, then $Z\restriction l_{j}\cap S$ must be contained in $S_{l_{j},1}$.

Recalling that $w^{\wedge}$ denotes $w\restriction l$, where $l$ is least such that $w(l)\neq 0$, we have

1. (d)

$|d^{S}_{m_{n}-1}|<|(w_{j}^{3})^{\wedge}|<\dots<|(w_{j}^{a_{j}})^{\wedge}|<|w_{j}^{3}|<\dots<|w_{j}^{a_{j}}|$. 2. (e)

If $|d^{S}_{m_{n}-1}|<l_{j+1}<l^{S}_{n}$, then $\max(l_{j},|w^{a_{j}}_{j}|)<|(w_{j+1}^{3})^{\wedge}|$.

For $k=3$, in the terminology of [14], (c) says that the only nodes in $S$ with which $\{w_{j}^{3},\dots,w_{j}^{a_{j}}\}$ has parallel $1$’s (pre-$3$-cliques) are in $S_{l_{j},1}$.

In what follows, we shall say that a strong coding tree $S$ such that $S\leq T$ is valid in $T$ if for each $m\in\mathbb{N}$, $r_{m}(S)$ is valid in $T$. Since $S$ is a strong coding tree, this is equivalent to $\max(r_{m}(S))$ being free in $T$ for each $m\in\mathbb{N}$.

Lemma 9.3**.**

Let $T\in\mathcal{T}_{k}$ be a strong coding tree. Then there is an incremental strong coding tree $S\leq T$ and a set of coding nodes $W\subseteq T$ such that each new pre-clique in $S$ is incrementally witnessed in $T$ by coding nodes in $W$.

Proof.

Recall that for any tree $T\in\mathcal{T}_{k}$, the sequence $\langle m_{n}:n\in\mathbb{N}\rangle$ denotes the indices such that $d^{T}_{m_{n}}=c^{T}_{n}$; that is, the $m_{n}$-th critical node in $T$ is the $n$-th coding node in $T$.

If $k=3$, fix some $S_{0}\in r_{m_{0}+1}[0,T]$ which is valid in $T$. Then $S_{0}$ has exactly one coding node, $c^{S_{0}}_{0}$, and it has ghost coding node $c^{S_{0}}_{-1}$, which is the shortest splitting node in $S_{0}$. There are no pre-cliques in $S_{0}$.

For $k=3$ and $n\geq 1$, or for $k\geq 4$ and $n\geq 0$, proceed as follows: If $k\geq 4$, let $S_{-1}$ consist of the stem of $T$, that is $d^{T}_{0}$. Suppose that we have chosen $S_{n-1}\in r_{m_{n-1}+1}[0,T]$ valid in $T$ and $W_{n-1}\subseteq T$ so that $S_{n-1}$ is incremental and each new pre-clique in $S_{n-1}$ is incrementally witnessed by some coding nodes in $W_{n-1}$. Take some $U_{n}\in r_{m_{n}+1}[S_{n-1},T]$ such that $r_{m_{n}}(U_{n})$ is valid in $T$. Let $V=\max(r_{m_{n}}(U_{n}))$.

Let $\langle X_{j}:j<\tilde{j}\rangle$ enumerate those subsets of $\max(U_{n})$ which have new pre-cliques over $r_{m_{n}}(U_{n})$ so that for each pair $j<j^{\prime}<\tilde{j}$, if $X_{j}$ is a new pre-$a$-clique and $X_{j^{\prime}}$ is a new pre-$a^{\prime}$-clique, then

1. (1)

$a\leq a^{\prime}$; 2. (2)

If $a=a^{\prime}$, then $X_{j}\not\supseteq X_{j^{\prime}}$.

Note that (1) implies that, in the case that $k\geq 4$, for each $a\in[3,k-1]$, all new pre-$a$-cliques are enumerated before any new pre-$(a+1)$-clique is enumerated. Furthermore, every new pre-clique in $\max(U_{n})$ over $r_{k_{n}}(U_{n})$ is enumerated in $\langle X_{j}:j<\tilde{j}\rangle$ whether or not it is maximal. By (2), all new pre-$a$-cliques composed of two nodes are listed before any new pre-$a$-clique consisting of three nodes, etc. For each $j<\tilde{j}$, let $Y_{j}=X_{j}\restriction(l_{V}+1)$.

By properties (1) and (2), $X_{0}$ must be a pre-$3$-clique consisting of two nodes. The construction process in this case is similar to the construction above for $S_{0}$ when $k>3$. By Lemma 6.10, there is a splitting node $s\in T$ such that $s$ is a sequence of [math]’s and $|s|>l_{V}+1$. Extend all nodes in $V$ leftmost in $T$ to the length $|s|$, and call this set of nodes $Z$. Apply Lemma 6.14 to obtain $V_{0}$ end-extending $Z$ so that the following hold: The node in $V_{0}$ extending $s^{\frown}1$ is a coding node, call it $w_{n,0}$; the two nodes in $V_{0}$ extending the nodes in $Y_{0}$ both have passing number $1$ at $w_{n,0}$; all other nodes in $V_{0}$ are leftmost extensions of the nodes in $V^{+}\setminus Y_{0}$; and the only new pre-clique in $V_{0}$ is the nodes in $V_{0}$ extending $Y_{0}$. Let $W_{n,0}=\{w_{n,0}\}$.

Given $j<\tilde{j}-1$ and $V_{j}$, let $Y^{\prime}_{j+1}$ be the set of those nodes in $V_{j}$ which extend the nodes in $Y_{j+1}$. Let $a\in[3,k]$ be such that $X_{j+1}$ is a new pre-$a$-clique. Applying Lemma 6.10 $a-2$ times, obtain splitting nodes $s_{i}$, $i<a-2$, in $T$ which are sequences of [math] such that $l_{V_{j}}<|s_{0}|<\dots<|s_{a-3}|$. Extend all nodes ${s_{i}}^{\frown}1$, $i<a-2$, leftmost in $T$ to length $|s_{a-3}|+1$; and extend the nodes in $V_{j}$ leftmost in $T$ to length $|s_{a-3}|+1$ and denote this set of nodes as $Z$. By Lemma 6.9, this adds no new pre-cliques over $V_{j}$. Next apply Lemma 6.14 $a-2$ times to obtain $V_{j+1}$ end-extending $Z$ and coding nodes $w_{n,j+1,i}\in T$, $i<a-2$, such that letting $Y^{\prime\prime}_{j+1}$ be those nodes in $V_{j+1}$ extending nodes in $Y^{\prime}_{j+1}$, the following hold:

1. (1)

$|w_{n,j+1,0}|<\dots<|w_{n,j+1,a-3}|$; 2. (2)

The nodes in $V_{j+1}$ all have length $|w_{n,j+1,a-3}|$; 3. (3)

For each $i<a-2$, all nodes in $\{w_{n,j+1,i^{\prime}}:i<i^{\prime}<a-2\}\cup Y^{\prime\prime}_{j+1}$ have passing number $1$ at $w_{n,j+1,i}$. 4. (4)

All nodes in $V_{j+1}\setminus Y^{\prime\prime}_{j+1}$ are leftmost extensions of nodes in $V_{j}\setminus Y^{\prime}_{j+1}$. 5. (5)

The only new pre-clique in $V_{j+1}$ above $V^{+}$ is the set of nodes in $Y^{\prime\prime}_{j+1}$.

Let $W_{n,j+1}=\{w_{n,j+1,i}:i<a-2\}$.

After $V_{\tilde{j}-1}$ has been constructed, take some $S_{n}\in r_{m_{n}+1}[r_{m_{n}}(U_{n}),T]$ such that $\max(S_{n})$ end-extends $V_{\tilde{j}-1}$, by Lemma 6.18. Let $W_{n}=\bigcup_{j<\tilde{j}}W_{n,j}$.

To finish, let $S=\bigcup_{n\in\mathbb{N}}S_{n}$ and $W=\bigcup_{n\in\mathbb{N}}W_{n}$. Then $S\leq T$, $S$ is incremental, and the pre-cliques in $S$ are incrementally witnessed by coding nodes in $W$. ∎

9.2. Ramsey theorem for strict similarity types

The main Ramsey theorem for strong coding trees is Theorem 9.9: Given a finite coloring of all strictly similar copies (Definition 9.4) of a fixed finite antichain in an incremental strong coding tree, there is a subtree which is again a strong coding tree in which all strictly similar copies of the antichain have the same color. Such antichains will have envelopes which have the Strict Witnessing Property. Moreover, all envelopes of a fixed incremental antichain of coding nodes will be strongly isomorphic to each other. This will allow for an application of Theorem 8.3 to obtain the same color for all copies of a given envelope, in some subtree in $\mathcal{T}_{k}$. From this, we will deduce Theorem 9.9.

Recall that a set of nodes $A$ is an antichain if no node in $A$ extends any other node in $A$. In what follows, by antichain, we mean an antichain of coding nodes. If $Z$ is an antichain, then the tree induced by $Z$ is the set of nodes

[TABLE]

We say that an antichain satisfies the Witnessing Property (Strict Witnessing Property) if and only if the tree it induces satisfies the Witnessing Property (Strict Witnessing Property).

Fix, for the rest of this section, an incremental strong coding tree $T\in\mathcal{T}_{k}$, as in Lemma 9.3. Notice that any strong coding subtree of $T$ will also be incremental. Furthermore, any antichain in $T$ must be incremental.

Definition 9.4** (Strict similarity type).**

Suppose $Z\subseteq T$ is a finite antichain of coding nodes. Enumerate the nodes of $Z$ in increasing order of length as $\langle z_{i}:i<\tilde{i}\rangle$ (excluding, as usual, new singleton pre-$3$-cliques). Enumerate all nodes in $Z^{\wedge}$ as $\langle u^{Z}_{m}:m<\tilde{m}\rangle$ in order of increasing length. Thus, each $u^{Z}_{m}$ is either a splitting node in $Z^{\wedge}$ or else a coding node in $Z$. List the minimal levels of new pre-cliques in $Z$ in increasing order as $\langle l_{j}:j<\tilde{j}\rangle$. For each $j<\tilde{j}$, let $I^{Z}_{l_{j}}$ denote the set of those $i<\tilde{i}$ such that $\{z_{i}\restriction l_{j}:i\in I^{Z}_{l_{j}}\}$ is the new pre-clique in $Z\restriction l_{j}$. The sequence

[TABLE]

is the strict similarity sequence of $Z$.

Let $Y$ be another finite antichain in $T$, and let

[TABLE]

be its strict similarity sequence. We say that $Y$ and $Z$ have the same strict similarity type or are strictly similar, and write $Y\stackrel{{\scriptstyle ss}}{{\sim}}Z$, if

1. (1)

The tree induced by $Y$ is strongly isomorphic to the tree induced by $Z$, so in particular, $\tilde{m}=\tilde{q}$; 2. (2)

$\tilde{j}=\tilde{k}$; 3. (3)

For each $j<\tilde{j}$, $I^{Y}_{p_{j}}=I^{Z}_{l_{j}}$; and 4. (4)

The function $\varphi:\{p_{j}:j<\tilde{j}\}\cup\{|u^{Y}_{m}|:m<\tilde{m}\}\rightarrow\{l_{j}:j<\tilde{j}\}\cup\{|u^{Z}_{m}|:m<\tilde{m}\}$, defined by $\varphi(p_{j})=l_{j}$ and $\varphi(u^{Y}_{m})=u^{Z}_{m}$, is an order preserving bijection between these two linearly ordered sets of natural numbers.

Define

[TABLE]

Note that if $Y\stackrel{{\scriptstyle ss}}{{\sim}}Z$, then the map $f:Y\rightarrow Z$ defined by $f(y_{i})=z_{i}$, for each $i<\tilde{i}$, induces the strong similarity map from the tree induced by $Y$ onto the tree induced by $Z$. Then $f(u^{Y}_{m})=u^{Z}_{m}$, for each $m<\tilde{m}$. Further, by (3) and (4) of Definition 9.4, this map preserves the order in which new pre-cliques appear, relative to all other new pre-cliques in $Y$ and $Z$ and the nodes in $Y^{\wedge}$ and $Z^{\wedge}$.

The following notion of envelope is defined in terms of structure without regard to an ambient strong coding tree. Given a fixed incremental strong coding tree $T$, in any given strong coding subtree $U\leq T$, there will certainly be finite subtrees of $U$ which have no envelope in $U$. The point of Lemma 9.3 is that there will be a strong coding subtree $S\leq U$ along with a set of witnessing coding nodes $W\subseteq U$ so that each finite antichain in $S$ has an envelope consisting of nodes from $W$. Thus, envelopes of antichains in $S$ will exist in $U$. Moreover, $S$ must be incremental, since $S\leq U\leq T$.

Definition 9.5** (Envelopes).**

Let $Z$ be a finite incremental antichain of coding nodes. An envelope of $Z$, denoted $E(Z)$, consists of $Z$ along with a set of coding nodes $W$ such that $Z\cup W$ satisfies Definition 9.2.

Thus, all new pre-cliques in an envelope $E(Z)=Z\cup W$ are incrementally witnessed by coding nodes in $W$. The set $W$ is called the set of witnessing coding nodes in the envelope. The next fact follows immediately from the definitions.

Fact 9.6**.**

Let $Z$ be any antichain in an incremental strong coding tree. Then any envelope of $Z$ has incrementally witnessed pre-cliques, which implies that $Z$ has the Strict Witnessing Property.

Lemma 9.7**.**

Let $Y$ and $Z$ be strictly similar incremental antichains of coding nodes. Then any envelope of $Y$ is strongly isomorphic to any envelope of $Z$, and both envelopes have the Strict Witnessing Property.

Proof.

Let $Y=\{y_{i}:i<\tilde{i}\}$ and $Z=\{z_{i}:i<\tilde{i}\}$ be the enumerations of $Y$ and $Z$ in order of increasing length, and let

[TABLE]

and

[TABLE]

be their strict similarity sequences, respectively. Let $E=Y\cup V$ and $F=Z\cup W$ be any envelopes of $Y$ and $Z$, respectively. For each $j<\tilde{j}$, let $a_{j}\geq 3$ be such that $I_{l_{j}}^{Y}$ is a new pre-$a_{j}$-clique. Then the members of $V$ may be labeled as $\{v_{j}^{3},\dots,v_{j}^{a_{j}}:j<\tilde{j}\}$ with the property that for each $j<\tilde{j}$, given the least $m<\tilde{m}$ such that $|v_{j}^{a_{j}}|<|u^{Y}_{m}|$, we have $|u^{Y}_{m-1}|<|(v_{j}^{3})^{\wedge}|$. This follows from Definition 9.2. Since $Y$ and $Z$ have the same strict similarity type, it follows that for each $j<\tilde{j}$, $I_{p_{j}}^{Z}$ is also a new pre-$a_{j}$-clique. Furthermore, $W=\{w_{j}^{3},\dots,w_{j}^{a_{j}}:j<\tilde{j}\}$, where for each $j<\tilde{j}$, given the least $m<\tilde{m}$ such that $|w_{j}^{a_{j}}|<|u^{Z}_{m}|$, we have that $|u^{Z}_{m-1}|<|(w_{j}^{3})^{\wedge}|$. Thus, $V$ and $W$ both have the same size, label it $J$.

Let $\tilde{n}=\tilde{i}+J$, and let $\{e_{n}:n<\tilde{n}\}$ and $\{f_{n}:n<\tilde{n}\}$ be the enumerations of $E$ and $F$ in order of increasing length, respectively. For each $j<\tilde{j}$, let $n_{j}$ be the index in $\tilde{n}$ such that $e_{n_{j}}=v_{j}$ and $f_{n_{j}}=w_{j}$. For $n<\tilde{n}$, let $E(n)$ denote the tree induced by $E$ restricted to those nodes of length less than or equal to $|e_{n}|$; precisely, $E(n)=\{e\restriction|t|:e\in E$, $t\in E^{\wedge}$, and $|t|\leq|e_{n}|\}$. Define $F(n)$ similarly.

We prove that $E\cong F$ by induction on $\tilde{j}$. If $\tilde{j}=0$, then $E=Y$ and $F=Z$, so $E\cong F$ follows from $Y\stackrel{{\scriptstyle ss}}{{\sim}}Z$. Suppose now that $\tilde{j}\geq 1$ and that, letting $j=\tilde{j}-1$, the induction hypothesis gives that $E(n)\cong F(n)$ for the maximal $n<\tilde{n}$ such that $e_{n}\in Y^{\wedge}$ and $|e_{n}|<l_{j}$. Let $m$ be the least integer below $\tilde{m}$ such that $|u^{Y}_{m}|>l_{j}$. Then $e_{n}=u^{Y}_{m-1}$ and the only nodes in $E^{\wedge}$ in the interval $(|u^{Y}_{m-1}|,|u^{Y}_{m}|)$ are $(v^{3}_{j})^{\wedge},\dots,(v^{a_{j}}_{j})^{\wedge},v^{3}_{j},\dots,v^{a_{j}}_{j}$. Likewise, the only nodes in $F^{\wedge}$ in the interval $(|u^{Z}_{m-1}|,|u^{Z}_{m}|)$ are $(w^{3}_{j})^{\wedge},\dots,(w^{a_{j}}_{j})^{\wedge},w^{3}_{j},\dots,w^{a_{j}}_{j}$.

By the induction hypothesis, there is a strong isomorphism $g:E(n)\rightarrow F(n)$. Extend it to a strong isomorphism $g^{*}:E(n^{\prime})\rightarrow F(n^{\prime})$, where $n^{\prime}=\tilde{n}-1$ as follows: Define $g^{*}=g$ on $E(n)$. For each $i\in[3,a_{j}]$, let $g^{*}((v^{i}_{j})^{\wedge})=(w^{i}_{j})^{\wedge}$ and $g^{*}(v^{i}_{j})=w^{i}_{j}$. Recall that the nodes $\{v^{3}_{j},\dots v^{a_{j}}_{j}\}$ form a pre-$(a_{j}-1)$-clique and only have mutual pre-cliques with nodes in $\{y_{i}:i\in I^{Y}_{l_{j}}\}$, witnessing this set, and no other members of $E$. Likewise, for $\{w^{3}_{j},\dots v^{a_{j}}_{j}\}$ and $\{z_{i}:i\in I^{Z}_{lp_{j}}\}$. Thus, $g^{*}$ from $E(n^{\prime\prime})$ to $F(n^{\prime\prime})$ is a strict similarity map, where $n^{\prime\prime}<\tilde{n}$ is the index such that $v^{a_{j}}_{j}=e_{n^{\prime\prime}}$. If $n^{\prime\prime}<\tilde{n}-1$, then $\{e_{q}:n^{\prime\prime}<q<\tilde{n}\}\subseteq Y^{\wedge}$ and $\{f_{q}:n^{\prime\prime}<q<\tilde{n}\}\subseteq Z^{\wedge}$. Since these sets have no new pre-cliques and are strictly similar, the map $g^{*}(e_{q})=f_{q}$, $n^{\prime\prime}<q<\tilde{n}$, is a strong isomorphism. Thus, we have constructed a strong isomorphism $g^{*}:E\rightarrow F$. It follows from the definitions that envelopes satisfy the Strict Witnessing Property. ∎

Lemma 9.8**.**

Suppose $Z$ is a finite antichain of coding nodes and $E$ is an envelope of $Z$ in $T$. Enumerate the nodes in $Z$ and $E$ in order of increasing length as $\langle z_{i}:i<\tilde{i}\rangle$ and $\langle e_{k}:k<\tilde{k}\rangle$, respectively. Given any $F\subseteq T$ with $F\cong E$, let $F\restriction Z:=\{f_{k_{i}}:i<\tilde{i}\}$, where $\langle f_{k}:k<\tilde{k}\rangle$ enumerates the nodes in $F$ in order of increasing length and for each $i<\tilde{i}$, $k_{i}$ is the index such that $e_{k_{i}}=z_{i}$. Then $F\restriction Z$ is strictly similar to $Z$.

Proof.

Recall that $E$ has incrementally witnessed new pre-cliques and $F\cong E$ implies that $F$ also has this property, and hence has the SWP. Let $\iota_{Z,F}:Z\rightarrow F$ be the injective map defined via $\iota_{Z,F}(z_{i})=f_{k_{i}}$, $i<\tilde{i}$, and let $F\restriction Z$ denote $\{f_{k_{i}}:i<\tilde{i}\}$, the image of $\iota_{Z,F}$. Then $F\restriction Z$ is a subset of $F$ which we claim is strictly similar to $Z$.

Since $F$ and $E$ each have incrementally witnessed new pre-cliques, the strong similarity map $g:E\rightarrow F$ satisfies that for each $j<\tilde{k}$, the indices of the new pre-cliques at level of the $j$-th coding node are the same:

[TABLE]

Since $\iota_{Z,F}$ is the restriction of $g$ to $Z$, $\iota_{Z,F}$ also takes each new pre-clique in $Z$ to the corresponding new pre-clique in $F\restriction Z$, with the same set of indices. Thus, $\iota_{Z,F}$ witnesses that $F\restriction Z$ is strictly similar to $Z$. ∎

Theorem 9.9** (Ramsey Theorem for Strict Similarity Types).**

Let $Z$ be a finite antichain of coding nodes in an incremental strong coding tree $T$, and suppose $h$ colors all subsets of $T$ which are strictly similar to $Z$ into finitely many colors. Then there is an incremental strong coding tree $S\leq T$ such that all subsets of $S$ strictly similar to $Z$ have the same $h$ color.

Proof.

First, note that there is an envelope $E$ of a copy of $Z$ in $T$: By Lemma 9.3, there is an incremental strong coding tree $U\leq T$ and a set of coding nodes $V\subseteq T$ such that each $Y\subseteq U$ which is strictly similar to $Z$ has an envelope in $T$ by adding nodes from $V$. Since $U$ is strongly isomorphic to $T$, there is subset $Y$ of $U$ which is strictly similar to $Z$. Let $E$ be any envelope of $Y$ in $T$, using witnessing coding nodes from $V$.

By Lemma 9.7, all envelopes of copies of $Z$ are strongly isomorphic and have the SWP. For each $F\cong E$, define $h^{*}(F)=h(F\restriction Z)$, where $F\restriction Z$ is the subset of $F$ provided by Lemma 9.8. The set $F\restriction Z$ is strictly similar to $Z$, so the coloring $h^{*}$ is well-defined. By Theorem 8.3, there is a strong coding tree $T^{\prime}\leq T$ such that $h^{*}$ is monochromatic on all strongly isomorphic copies of $E$ in $T^{\prime}$. Lemma 9.3 implies there is an incremental strong coding tree $S\leq T^{\prime}$ and a set of coding nodes $W\subseteq T^{\prime}$ such that each $Y\subseteq S$ which is strictly similar to $Z$ has an envelope $F$ in $T^{\prime}$, so that $h(Y)=h^{*}(F)$. Therefore, $h$ takes only one color on all strictly similar copies of $Z$ in $S$. ∎

10. The Henson graphs have finite big Ramsey degrees

From the results in previous sections, we now prove the main theorem of this paper, Theorem 10.2. This result follows from Ramsey Theorem 9.9 for strict similarity types along with Lemma 10.1 below.

For a strong coding tree $T$, let $(T,\subseteq)$ be the reduct of $(T,\mathbb{N};\subseteq,<,c)$. Then $(T,\subseteq)$ is simply the tree structure of $T$, disregarding the difference between coding nodes and non-coding nodes. We say that two trees $(T,\subseteq)$ and $(S,\subseteq)$ are strongly similar trees if they satisfy Definition 3.1 in [52]. This is the same as modifying Definition 5.3 by deleting (6) and changing (7) to apply to passing numbers of all nodes in the trees. By saying that two finite trees are strongly similar trees, we are implicitly assuming that their extensions to their immediate successors of their maximal nodes are still strongly similar. Thus, strong similarity of finite trees implies passing numbers of their immediate extensions are preserved. Given an antichain $D$ of coding nodes from a strong coding tree, let $L_{D}$ denote the set of all lengths of nodes $t\in D^{\wedge}$ such that $t$ is not the splitting predecessor of any coding node in $D$. Define

[TABLE]

Then $(D^{*},\subseteq)$ is a tree.

Lemma 10.1**.**

Let $T\in\mathcal{T}_{k}$ be a strong coding tree. Then there is an infinite antichain of coding nodes $D\subseteq T$ which code $\mathcal{H}_{k}$ in the same way as $\mathbb{T}_{k}$: $c^{D}_{n}(l^{D}_{i})=c^{k}_{n}(l^{k}_{i})$, for all $i<n$. Moreover, $(D^{*},\subseteq)$ and $(\mathbb{T}_{k},\subseteq)$ are strongly similar as trees.

Proof.

We will construct a subtree $\mathbb{D}\subseteq\mathbb{T}_{k}$ such that $D$ the set of coding nodes in $\mathbb{D}$ form an antichain satisfying the lemma. Then, since $T\in\mathcal{T}_{k}$ implies $T\cong\mathbb{T}_{k}$, letting $\varphi:\mathbb{T}_{k}\rightarrow T$ be the strong similarity map between $\mathbb{T}_{k}$ and $T$, the image of $\varphi$ on the coding nodes of $\mathbb{D}$ will yield an antichain of coding nodes $D\subseteq T$ satisfying the lemma.

We will construct $\mathbb{D}$ so that for each $n$, the node of length $l^{\mathbb{D}}_{n}$ which is going to be extended to the next coding node $c^{\mathbb{D}}_{n+1}$ will extend to a splitting node in $\mathbb{D}$ of length smaller than that of any other splitting node in the $(n+1)$-st interval of $\mathbb{D}$. Above that splitting node, the splitting will be regular in the interval until the next coding node. Recall that for each $i\in\mathbb{N}$, $\mathbb{T}_{k}$ has either a coding node or else a splitting node of length $i$. To avoid some superscripts, let $l_{n}=|c^{k}_{n}|$ and $p_{n}=|c^{\mathbb{D}}_{n}|$. Let $j_{n}$ be the index such that $c^{\mathbb{D}}_{n}=c^{k}_{j_{n}}$, so that $p_{n}$ equals $l_{j_{n}}$. The set of nodes in $\mathbb{D}\setminus\{c^{\mathbb{D}}_{n}\}$ of length $p_{n}$ shall be indexed as $\{d_{t}:t\in\mathbb{T}_{k}\cap\mathcal{S}_{l_{n}}\}$. Recall that $m_{n}$ is the index such that the $m_{n}$-th critical node $d^{k}_{m_{n}}$ of $\mathbb{T}_{k}$ is the $n$-th coding node $c^{k}_{n}$ of $\mathbb{T}_{k}$.

We define inductively on $n\geq-1$ finite trees with coding nodes, $r_{m_{n}+1}(\mathbb{D}):=\mathbb{D}\cap\mathcal{S}_{\leq p_{n}}$, and strong similarity maps of the trees $\varphi:r_{m_{n}+1}(\mathbb{T}_{k})\rightarrow r_{m_{n}+1}(\mathbb{D}^{*})$, where $|\varphi(c^{k}_{n})|=p_{n}$. Recall that the node $\langle\rangle$ is the ghost coding node $c^{k}_{-1}$ in $\mathbb{T}_{k}$. Define $d_{\langle\rangle}=\varphi(\langle\rangle)=\langle\rangle$. The node $\langle\rangle$ splits in $\mathbb{T}_{k}$, so the node $d_{\langle\rangle}$ will split in $\mathbb{D}$. Suppose that $n\in\mathbb{N}$ and we have constructed $r_{m_{n-1}+1}(\mathbb{D})$ satisfying the lemma. By the induction hypothesis, there is a strong similarity map of the trees $\varphi:r_{m_{n-1}+1}(\mathbb{T}_{k})\rightarrow r_{m_{n-1}+1}(\mathbb{D}^{*})$. For $t\in\mathbb{T}_{k}(m_{n-1})$, let $d_{t}$ denote $\varphi(t)$.

Let $s$ denote the node in $\mathbb{T}_{k}(m_{n-1})$ which extends to the coding node $c^{k}_{n}$. Let $v_{s}$ be a splitting node in $\mathbb{T}_{k}$ extending $d_{s}$. Let $u_{s}={v_{s}}^{\frown}1$ and extend all nodes $d_{t}$, $t\in\mathbb{T}_{k}(m_{n-1})\setminus\{s\}$, leftmost to length $|u_{s}|$ and label these $d_{t}^{\prime}$. Extend ${v_{s}}^{\frown}0$ leftmost to length $|u_{s}|$ and label it $d^{\prime}_{s}$. Let $X=\{d^{\prime}_{t}:t\in\mathbb{T}_{k}(m_{n-1})\}\setminus\{u_{s}\}$ and let

[TABLE]

Apply Lemma 6.15 to obtain a coding node $c^{\mathbb{D}}_{n}$ extending $u_{s}$ and nodes $d_{w}$, $w\in\mathbb{T}_{k}(m_{n})$, so that, letting $p_{n}=|c^{\mathbb{D}}_{n}|$ and

[TABLE]

and for $m\in(m_{n-1},m_{n})$, defining $\mathbb{D}(m)=\{d_{t}\restriction|s_{m}|:t\in\mathbb{D}(m_{n})\}$, where $s_{m}$ is the $m$-th splitting node in $\mathbb{D}(m_{n})^{\wedge}$, the following hold: $r_{m_{n}+1}(\mathbb{D})$ satisfies the Witnessing Property and $r_{m_{n}+1}(\mathbb{D}^{*})$ is strongly similar as a tree to $r_{m_{n}+1}(\mathbb{T}_{k})$. Thus, the coding nodes in $r_{m_{n}+1}(\mathbb{D})$ code exactly the same graph as the coding nodes in $r_{m_{n}+1}(\mathbb{T}_{k})$.

Let $\mathbb{D}=\bigcup_{m\in\mathbb{N}}\mathbb{D}(m)$. Then the set of coding nodes in $\mathbb{D}$ forms an antichain of maximal nodes in $\mathbb{D}$. Further, the tree generated by the the meet closure of the set $\{c^{\mathbb{D}}_{n}:n<\omega\}$ is exactly $\mathbb{D}$, and $\mathbb{D}^{*}$ and $\mathbb{T}_{k}$ are strongly similar as trees. By the construction, for each pair $i<n<\omega$, $c^{\mathbb{D}}_{n}(p_{i})=c^{k}_{n}(l_{i})$; hence they code $\mathcal{H}_{k}$ in the same order.

To finish, let $f_{T}$ be the strong isomorphism from $\mathbb{T}_{k}$ to $T$. Letting $D$ be the $f_{T}$-image of $\{c^{\mathbb{D}}_{n}:n<\omega\}$, we see that $D$ is an antichain of coding nodes in $T$ such that $D^{*}$ and $\mathbb{D}^{*}$ are strongly similar trees, and hence $D^{*}$ is strongly similar as a tree to $\mathbb{T}_{k}$. Thus, the antichain of coding nodes $D$ codes $\mathcal{H}_{k}$ and satisfies the lemma. ∎

Recall that the Henson graph $\mathcal{H}_{k}$ is, up to isomorphism, the homogeneous $k$-clique-free graph on countably many vertices which is universal for all $k$-clique-free graphs on countably many vertices.

Main Theorem 10.2**.**

For each $k\geq 3$, the Henson graph $\mathcal{H}_{k}$ has finite big Ramsey degrees.

Proof.

Fix $k\geq 3$ and let $\mathrm{G}$ be a finite $K_{k}$-free graph. Suppose $f$ colors of all the copies of $\mathrm{G}$ in $\mathcal{H}_{k}$ into finitely many colors. By Example 5.1, there is a strong coding tree $\mathbb{T}_{k}$ such that the coding nodes in $\mathbb{T}_{k}$ code a $\mathcal{H}_{k}$. Let $\mathcal{A}$ denote the set of all antichains of coding nodes of $\mathbb{T}_{k}$ which code a copy of $\mathrm{G}$. For each $Y\in\mathcal{A}$, let $h(Y)=f(\mathrm{G}^{\prime})$, where $\mathrm{G}^{\prime}$ is the copy of $\mathrm{G}$ coded by the coding nodes in $Y$. Then $h$ is a finite coloring on $\mathcal{A}$.

Let $n(\mathrm{G})$ be the number of different strict similarity types of incremental antichains of coding nodes in of $\mathbb{T}_{k}$ coding $\mathrm{G}$, and let $\{Z_{i}:i<n(\mathrm{G})\}$ be a set of one representative from each of these strict similarity types. Successively apply Theorem 9.9 to obtain incremental strong coding trees $\mathbb{T}_{k}\geq T_{0}\geq\dots\geq T_{n(\mathrm{G})-1}$ so that for each $i<n(\mathrm{G})$, $h$ is takes only one color on the set of incremental antichains of coding nodes $A\subseteq T_{i}$ such that $A$ is strictly similar to $Z_{i}$. Let $S=T_{n(\mathrm{G})-1}$.

By Lemma 10.1 there is an antichain of coding nodes $D\subseteq S$ which codes $\mathcal{H}_{k}$ in the same way as $\mathbb{T}_{k}$. Every set of coding nodes in $D$ coding $\mathrm{G}$ is automatically incremental, since $S$ is incremental. Therefore, every copy of $\mathrm{G}$ in the copy of $\mathcal{H}_{k}$ coded by the coding nodes in $\mathrm{D}$ is coded by an incremental antichain of coding nodes. Thus, the number of strict similarity types of incremental antichains in $\mathbb{T}_{k}$ coding $\mathrm{G}$ provides an upper bound for the big Ramsey degree of $\mathrm{G}$ in $\mathcal{H}_{k}$. ∎

Thus, each Henson graph has finite big Ramsey degrees. Moreover, given a finite $k$-clique-free graph, the big Ramsey degree $T(\mathrm{G},\mathcal{H}_{k})$ is bounded by the number of strict similarity types of incremental antichains coding copies of $\mathrm{G}$ in $\mathbb{T}_{k}$.

11. Future Directions

This article developed a unified approach to proving upper bounds for big Ramsey degrees of all Henson graphs. The main phases of the proof were as follows: I. Find the correct structures to code $\mathcal{H}_{k}$ and prove Extension Lemmas. II. Prove an analogue of Milliken’s Theorem for finite trees with certain structure. In the case of the Henson graphs, this is the Strict Witnessing Property. III. Find a means for turning finite antichains into finite trees with the Strict Witnessing Property so as to deduce a Ramsey Theorem for finite antichains from the previous Milliken-style theorem. This general approach should apply to a large class of ultrahomogeneous structures with forbidden configurations. It will be interesting to see where the dividing line is between those structures for which this methodology works and those for which it does not. The author conjectures that similar approaches will work for forbidden configurations which are irreducible in the sense of [43] and [44].

Although we have not yet proved the lower bounds to obtain the precise big Ramsey degrees $T(\mathrm{G},\mathcal{K}_{k})$, we conjecture that they will be exactly the number of strict similarity types of incremental antichains coding $\mathrm{G}$. We further conjecture that once found, the lower bounds will satisfy the conditions needed for Zucker’s work in [57]. If so, then each Henson graph would admit a big Ramsey structure and any big Ramsey flow will be a universal completion flow, and any two universal completion flows will be universal. We mention some bounds on big Ramsey degrees, found by computing the number of distinct strict similarity types of incremental antichains of coding nodes. Let $\bar{K}_{2}$ denote two vertices with no edge between them. $T(K_{2},\mathcal{H}_{3})=2$ was proved by Sauer in [51]. This is in fact the number of strict similarity types of two coding nodes coding an edge in $\mathbb{T}_{3}$. The number of strict similarity types of incremental antichains coding a non-edge in $\mathbb{T}_{3}$ is seven, so

[TABLE]

The number of strict similarity types of incremental antichains coding an edge in $\mathbb{T}_{4}$ is 44, so

[TABLE]

The number of strict similarity types of incremental antichains coding a non-edge in $\mathbb{T}_{4}$ is quite large. These numbers grow quickly as more pre-cliques are allowed. Since any copy of $\mathcal{H}_{k}$ can be enumerated and coded by a strong coding tree, which wlog can be assumed to be incremental, it seems that these strict similarity types should persist.

We point out that by a compactness argument, one can obtain finite versions of the two main Ramsey theorems in this article. In particular, the finite version of Theorem 9.9 may well produce better bounds for the sizes of finite $K_{k}$-free graphs instantiating that the Fraïssé  class $\mathcal{G}_{k}^{<}$ has the Ramsey property.

Curiously, the methodology in this paper and [14] is also having impact on Fraïssé structures without forbidden configurations. In [10], the author recently developed trees with coding nodes to code copies of the Rado graph and used forcing arguments similar to, but much simpler than, those in Section 7 to answer a question of [34] regarding infinite dimensional Ramsey theory of copies of the Rado graph. These methods work also for the rationals, and ongoing work is to discover all aspects of Ramsey and anti-Ramsey theorems for colorings of definable sets of spaces of such Fraïssé structures, the aim being to extend theorems of Galvin-Prikry in [25] and Ellentuck in [21] to a wide collection of Fraïssé classes. Lastly, modifications and generalizations of this approach seem likely to produce a general theorem for big Ramsey degrees for a large collection of relational Fraïssé structures without forbidden configurations.

11.1. Updates: May, 2022

Since the posting of the first version of this paper on the arxiv in January 2019, much progress has been made on big Ramsey degrees and related Ramsey theory of infinite structures. The author’s paper [11], which will appear in the Proceedings of the 2022 International Congress of Mathematicians, provides an overview of current knowledge of big Ramsey degrees of countable homogeneous structures. Here, we briefly mention recent work which has been directly influenced by this paper and its predecessor [14], which found upper bounds for big Ramsey degress of the triangle-free Henson graph, $\mathcal{H}_{3}$.

Via a small tweak of the strong coding trees in [14], exact big Ramsey degrees for $\mathcal{H}_{3}$ were characterized by the author in [12]. (The small tweak is an instance of the more general notion of “controlled coding levels”–Definition 5.6.5 of [3].) An equivalent characterization (based on upper bounds in [33] and [58]) was found independently around the same time by Balko, Chodounský, Hubička, Konečný, Vena, and Zucker in an unposted preprint. Big Ramsey degrees in $\mathcal{H}_{3}$ are exactly characterized by diagonal antichains of coding nodes in the strong coding trees defined in [12] and the order in which pairs of new pre-$3$-cliques appear. This is the part of the author’s notion of strict similarity type essential to unavoidable colorings. In particular, the big Ramsey degree $T(\bar{K}_{2},\mathcal{H}_{3})$ is now known to be exactly 5.

Building on methods in this paper, Zucker in [58] developed an approach using fully branching (similar to our Definition 4.1) “left-leaning” coding trees, forcing, and a new style of envelope to handle all free amalgamation classes with finitely many relations of arity at most two and finitely many forbidden irreducible structures; he proved that their Fraïssé limits have finite big Ramsey degrees. Zucker crystalized the essential properties of new pre-cliques in this paper to so-called ‘age-changes’ which made possible his broad approach. The use of fully branching coding trees led to a shorter proof of upper bounds, though at the cost of looser bounds for Henson graphs than those in this paper. By combining Zucker’s approach in [58] with properties inherent in the strong coding trees and strict similarity types in this paper as well as new ideas, exact big Ramsey degrees were characterized for all free amalgamation classes with finitely many relations of arity at most two and finitely many forbidden irreducible structures in the recent paper [3] by Balko, Chodounský, Dobrinen, Hubička, Konečný, Vena, and Zucker. In particular, we now know that the big Ramsey degree $T(K_{2},\mathcal{H}_{4})$ is exactly 17.

Building on ideas from this paper in another direction is the work of Coulson, Dobrinen, and Patel in [8]. That work distills an amalgamation property true for the rationals and the Rado graph, and false for Henson graphs, called SDAP, which guarantees that the forcing partial order of the sort used in Theorem 7.2 here can be simply end-extension. By working with diagonal coding trees for Fraïssé structures satisfying a strengthened version of SDAP, called SDAP+, the following results were obtained: (1) Indivisibility for Fraïssé limits satisfying SDAP+ with finitely many relations of finite arity; (2) Exact big Ramsey degrees for Fraïssé limits satisfying SDAP+ with finitely many relations of arity at most two, moreover characterized just by diagonal antichains and passing types. Some interesting byproducts of the approach in [8] are that exact big Ramsey degrees are found directly, with no need for envelopes, and previous results [36], [37], and [52] are recovered by the one method.

We point out that the exact big Ramsey degrees in [3] and in [8] satisfy the conditions of Zucker in [57] to admit big Ramsey structures.

A third line of results using coding trees involve infinite dimensional Ramsey theory for Fraïssé structures. The goal is to extend the Galvin-Prikry [25] or Ellentuck [21] theorem to subspaces of the Baire space in which all points (infinite subsets of $\omega$) represent a subcopy of a given Fraïssé structure; coding trees help ensure that one is actually taking a subcopy of the given Fraïssé structure, not just any infinite substructure. An investigation of this area was asked for in [34] and begun in the author’s paper [10], which obtained a Galvin-Prikry analogue for the Rado graph. The author has extended this to Fraïssé structures satisfying SDAP+ with finitely many relations of arity at most two, building on work in [8]; the Galvin-Prikry analogue there recovers exact big Ramsey degrees. Work in this direction continues.

In 2020, Hubička found a new way to obtain upper bounds for big Ramsey degrees of the generic partial order as well as the triangle-free Henson graph in [33]. His method used parameter spaces of words and a coding of pre-$3$-cliques in that setting, producing the first forcing-free proof of upper bounds for $\mathcal{H}_{3}$. This formed the basis for the characterization of exact big Ramsey degrees of the generic partial order in [2], by Balko, Chodounský, Dobrinen, Hubička, Konečný, Vena, and Zucker. Hubička’s method for upper bounds has been extended to upper bounds for relational structures in a finite relational language with relations of arity at most 2 described by forbidden induced cycles in [4]. Big Ramsey degrees for structures with relations of higher arities was initiated in [5] and [6] for the generic $3$-regular hypergraph, using the product Milliken theorem. Category-theoretic methods are also currently being employed to discover exact big Ramsey degrees as well as transport existence of upper bounds in the work of Dasilva Barbosa [7] and Mašulović in [40]. Work on further types of structures with an expanding collection of methods is continuing to be fruitful.