More on tree properties

TL;DR

This paper explores the properties of tree properties in model theory, providing criteria for SOP$_2$ and SOP$_1$, and examining the relationship between TP$_2$ and Kim-forking, with implications for supersimplicity.

Contribution

It introduces type-counting criteria for SOP$_2$ and SOP$_1$, and investigates the connection between TP$_2$ and Kim-forking, advancing understanding of independence notions.

Findings

Type-counting criteria for SOP$_2$ and SOP$_1$ are established.

A theory is supersimple iff there are no countably infinite Kim-forking chains.

Relationships between TP$_2$ and Kim-forking are clarified.

Abstract

Tree properties are introduced by Shelah, and it is well-known that a theory has TP (the tree property) if and only if it has TP$_1$ or TP$_2$. In any simple theory (i.e., a theory not having TP), forking supplies a good independence notion as it satisfies symmetry, transitivity, extension, local character, and type-amalgamation. Shelah also introduced SOP$_n$ ($n$-strong order property). Recently it is proved that in any NSOP$_1$ theory (i.e. a theory not having SOP$_1$) holding nonforking existence, Kim-forking also satisfies all the mentioned independence properties except base monotonicity (one direction of transitivity). These results are the sources of motivation for this paper. Mainly, we produce type-counting criteria for SOP$_2$ (which is equivalent to TP$_1$) and SOP$_1$. In addition, we study relationships between TP$_2$ and Kim-forking, and obtain that a theory is supersimple iff there is no countably infinite Kim-forking chain.

Equations

$$\mathcal{G}=\{q_{\delta }\}\cup \bigcup \{{\mathcal{G}}_{(\delta \lceil i{)}^{⌢}⟨j_i⟩}\mid i<\kappa ,j_i<\lambda ,j_i\ne \delta (i)\},$$

$$d{/⌣{}^{|}}_{A_i}^K{a}_{i0}^{\ast }\text{mbox}andd{/⌣{}^{|}}_{A_i}^K{A}_{i+1}$$

$$d{/⌣{}^{|}}_{E_i}^K{a}_{i0}^{\ast }\text{mbox}andhenced{/⌣{}^{|}}_{E_i}^K{E}_{i+1}.$$

Full text

Mainly, we produce type-counting criteria for SOP2 (which is equivalent to TP1) and SOP1. In addition, we study relationships between TP2 and Kim-forking, and obtain that a theory is supersimple iff there is no countably infinite Kim-forking chain.

The first author has been partially funded by a Spanish government grant MTM2017-86777-P and a Catalan DURSI grant 2017SGR-270. The second author has been supported by Samsung Science Technology Foundation under Project Number SSTF-BA1301-03 and an NRF of Korea grant 2018R1D1A1A02085584.

In this paper we study various notions of tree properties, and we mainly produce type-counting criteria for SOP1 and SOP2. TP (the tree property) is introduced by S. Shelah in [17], and it is shown that in any simple theory (a theory not having TP), forking satisfies local character, finite character, extension, and later in [10],[14], symmetry, full transitivity, and type-amalgamation of Lascar types, over arbitrary sets.

In [16], it is claimed that a theory has TP if and only if it has TP1 or TP2, and a complete proof is supplied in [13]. On the other hand, in [18], Shelah introduces the notions of $n$-strong order properties (SOPn) for $n\geq 3$, which further classify theories having TP1. More precisely, a theory has SOPn if there is a formula $\varphi(x,y)$ ($|x|=|y|$) defining a directed graph that has an infinite chain but no cycle of length $\leq n$. Hence SOPn+1 implies SOPn, but it is known that the implication is not reversible for each $n\geq 3$. As we are not dealing with SOPn for $n\geq 3$ in this note, we do not give many details on this.

For $n=1,2,$ Shelah defines SOPn separately as follows.

1. (1)

We say a formula $\varphi(x,y)$ has SOP2 if there is a set $\{a_{\alpha}\mid\alpha\in 2^{<\omega}\}$ of tuples such that

1. (a)

for each $\beta\in 2^{\omega}$, $\{\varphi(x,a_{\beta\lceil n})\mid n\in\omega\}$ is consistent, and 2. (b)

for each incomparable pair $\gamma,\gamma^{\prime}\in 2^{<\omega}$, $\{\varphi(x,a_{\gamma}),\varphi(x,a_{\gamma^{\prime}})\}$ is inconsistent.

A theory $T$ has SOP2 if some formula in $T$ has SOP2. 2. (2)

We say a formula $\varphi(x,y)$ has SOP1 if there is a set $\{a_{\alpha}\mid\alpha\in 2^{<\omega}\}$ of tuples such that

1. (a)

for each $\beta\in 2^{\omega}$, $\{\varphi(x,a_{\beta\lceil n})\mid n\in\omega\}$ is consistent, and 2. (b)

for each $\beta\in 2^{<\omega}$, $\{\varphi(x,a_{\gamma}),\varphi(x,a_{\beta^{\smallfrown}1})\}$ is inconsistent whenever $\beta^{\smallfrown}0\unlhd\gamma$.

A theory $T$ has SOP1 if some formula in $T$ has SOP1. We say a theory $T$ is NSOP1 if $T$ does not have SOP1.

Hence it follows that SOP2 implies SOP1. It is known for a theory that SOP3 implies SOP2, and SOP2 is equivalent to TP1. It is still an open question whether conversely, SOP1 implies SOP3, or SOP2. The random parametrized equivalence relations (Example 3.4), an infinite dimensional vector space over an algebraically closed field with a bilinear form, and $\omega$-free PAC fields are typical examples having non-simple but NSOP1 theories. Recently in [7],[8], it is shown that in any NSOP1 theory, over models, ‘Kim-forking’ satisfies all the aforementioned axioms that forking satisfies in simple theories, except base monotonicity (one direction of full transitivity). Then it is proved in [6], [4] that the same axioms hold over arbitrary sets in any NSOP1 theory having nonforking existence. So far summarized results justify our study of various tree properties in this paper.

Throughout this note, we use standard notation. We work in a large saturated model ${\mathcal{M}}$ of a complete theory $T$ in a language ${\mathcal{L}}$, and $a,b,\dots$ ($A,B,\dots$) denote finite (small, resp.) tuples (sets, resp.) from ${\mathcal{M}}$, unless said otherwise. We write $a\equiv_{A}b$ to mean $\operatorname{tp}(a/A)=\operatorname{tp}(b/A)$. As is customary, for cardinals $\kappa,\lambda$, we write $\lambda^{\kappa}$, $\lambda^{<\kappa}$ to denote $\{f\mid f:\kappa\to\lambda\}$, $\{f\mid f:\alpha\to\lambda,\ \alpha\in\kappa\}$ respectively, or their cardinalities, and it will be clear from context which one they mean. As usual, we can look at $\lambda^{<\kappa}=\{f\mid f:\alpha\to\lambda,\ \alpha<\kappa\}$ as a tree, and we give a partial order $\unlhd$ to it. Namely we let $\alpha\unlhd\beta$ for $\alpha,\beta\in\lambda^{<\kappa}$, when $\alpha=\beta\lceil\operatorname{dom}(\alpha)$. Thus we say $\alpha,\beta$ are incomparable if so are they in the ordering $\unlhd$. Also $\alpha^{\smallfrown}\beta$ denotes the concatenation of $\beta$ after $\alpha$. When $\beta=\langle i_{0},\dots,i_{n}\rangle$ where $i_{0},\dots,i_{n}\in\lambda$, we may simply write $\alpha i_{0}\cdots i_{n}$ to mean $\alpha^{\smallfrown}\beta$, so for example $\alpha^{\smallfrown}1$ or $\alpha 1$ indeed means $\alpha^{\smallfrown}\langle 1\rangle$. In this note if we write a set as $\{p_{i}\mid i\in I\}$ then $p_{i}\neq p_{j}$ for $i\neq j\in I$. Given a sequence of tuples $\langle c_{i}\mid i<\kappa\rangle$ and $j<\kappa$, we write $c_{<j}$, $c_{>j}$ to abbreviate $\langle c_{i}\mid i<j\rangle$, $\langle c_{i}\mid j<i<\kappa\rangle$, respectively.

We now state definitions and facts including those already mentioned that will be freely used throughout the paper.

1. (1)

We say an ${\mathcal{L}}$-formula $\varphi(x,y)$ has the $k$-tree property ($k$-TP) where $k\geq 2$, if there is the set of tuples $\{c_{\beta}\mid\beta\in\omega^{<\omega}\}$ (from ${\mathcal{M}}$) such that for each $\alpha\in\omega^{\omega}$, $\{\varphi(x,c_{\alpha\lceil n})\mid n\in\omega\}$ is consistent, while for any $\beta\in\omega^{<\omega}$, $\{\varphi(x,c_{\beta^{\smallfrown}i})\mid i\in\omega\}$ is $k$-inconsistent (i.e. any $k$-subset is inconsistent). A formula has the tree property (TP) if it has $k$-TP for some $k\geq 2$. We say $T$ has TP if a formula in $T$ has this property. We say $T$ is simple if $T$ does not have TP. 2. (2)

A formula $\psi(x,y)$ has the tree property of the first kind (TP1) if there are tuples $a_{\alpha}$ ($\alpha\in\omega^{<\omega}$) such that $\{\psi(x,a_{\beta\lceil n})|n\in\omega\}$ is consistent for each $\beta\in\omega^{\omega}$, while $\psi(x,a_{\alpha})\wedge\psi(x,a_{\gamma})$ is inconsistent whenever $\alpha,\gamma\in\omega^{<\omega}$ are incomparable. A theory has TP1 if so has a formula. 3. (3)

We say a formula $\psi(x,y)\in{\mathcal{L}}$ has the tree property of the second kind (TP2) if there are tuples $a^{i}_{j}$ ($i,j<\omega$) such that for each $i$, $\{\psi(x,a^{i}_{j})\mid j<\omega\}$ is 2-inconsistent, whereas for any $f\in\omega^{\omega}$, $\{\psi(x,a^{i}_{f(i)})\mid i<\omega\}$ is consistent. We say $T$ has TP2 if a formula has so in $T$.

1. (1)

The following are equivalent.

1. (a)

A theory $T$ has TP. 2. (b)

$T$* has $2$-TP.* 3. (c)

$T$* has either TP1 or TP2.* 2. (2)

A formula has TP1* iff it has SOP2.* 3. (3)

If a formula has SOP1* then it has $2$-TP.*

In Fact 0.3(1), the equivalence of (a) and (b) is shown in [16], and that of (a) and (c) is claimed in [16], but a correct proof is stated in [13]. Fact 0.3(2)(3) easily come from the definitions.

1. (1)

A formula $\varphi(x,y)\in{\mathcal{L}}$ has SOP1*.* 2. (2)

There is a sequence $\langle a_{i}b_{i}\mid i<\omega\rangle$ such that

1. (a)

$a_{i}\equiv_{(ab)_{<i}}b_{i}$* for all $i<\omega$,* 2. (b)

$\{\varphi(x,a_{i})\mid i<\omega\}$* is consistent, and* 3. (c)

$\{\varphi(x,b_{i})\mid i<\omega\}$* is $2$-inconsistent.*

In Section 1, we supply type-counting criteria for SOP2. These are generalizations of those in [12], and we use similar techniques in [1] where analogous criteria for TP are stated.

In Section 2, in parallel, we produce type-counting criteria for SOP1.

In Section 3, we study TP2 in relation with Kim-independence and local weights. In particular we show that $T$ is supersimple iff there is no Kim-forking chain of length $\omega$.

1. Type-counting criteria for SOP2

When Shelah introduces the class of simple theories in [17], he states and proves type-counting criteria for TP. Then in [1], the first author improves those and suggests more elaborate criteria for TP. Later in [12], type-counting criteria for TP1 (equivalently for SOP2) analogous to the type-counting results of [17] are suggested. In [15], another type-counting criteria for SOP2 is suggested. Now in this section, we supply more refined criteria for SOP2, which are analogous to those for TP in [1].

Definition 1.1**.**

Let $\varphi(x,y)$ be an ${\mathcal{L}}$-formula. Assume infinite cardinals $\kappa,\lambda$ are given. We define $\mathrm{NT}_{\varphi}^{2}(\kappa,\lambda)$ as the supremum of cardinalities $|\mathcal{F}|$ of sets $\mathcal{F}$ of positive $\varphi$-types $p(x)$ over some fixed set $A$ of cardinality $\lambda$ satisfying that

1. (1)

$|p(x)|=\kappa$ for every $p(x)\in\mathcal{F}$, and 2. (2)

for every subfamily $\{p_{i}\mid i<\lambda^{+}\}\subseteq\mathcal{F}$, there are disjoint subsets $\tau_{j}\subset\lambda^{+}$ with $|\tau_{j}|=\lambda^{+}$, and families $\{p^{\prime}_{i}\mid p^{\prime}_{i}\subseteq p_{i},\ i\in\tau_{j}\}$ $(j=0,1)$ such that $|p_{i}\smallsetminus p^{\prime}_{i}|<\kappa$ for each $i\in\tau_{0}\cup\tau_{1}$, and every formula in $\bigcup_{i\in\tau_{1}}p^{\prime}_{i}$ is inconsistent with every formula in $\bigcup_{i\in\tau_{0}}p^{\prime}_{i}$.

Notice that if $|{\mathcal{F}}|\leq\lambda$ then the condition (2) is vacuous.

We define $\mathrm{NT}^{2}(\kappa,\lambda)$ in a similar way, with the only difference that each partial type $p(x)\in\mathcal{F}$ (with finite $x$) may contain any formula over $A$, not only instances of a fixed $\varphi(x,y)$, while still $|p(x)|=\kappa$.

Now given a formula $\varphi$, we give type-counting criteria for SOP2, in terms of NT${}^{2}_{\varphi}$.

Theorem 1.2**.**

Let $\kappa,\lambda$ denote infinite cardinals. The following are equivalent for a formula $\varphi(x,y)\in{\mathcal{L}}$.

1. (1)

$\varphi(x,y)$* has SOP2.* 2. (2)

$\mathrm{NT}_{\varphi}^{2}(\omega,\omega)\geq\omega_{1}$** 3. (3)

$\mathrm{NT}_{\varphi}^{2}(\omega,\omega)\geq 2^{\omega}.$** 4. (4)

$\mathrm{NT}_{\varphi}^{2}(\kappa,\lambda)\geq\lambda^{+}$* for some $\kappa,\lambda$.* 5. (5)

$\mathrm{NT}_{\varphi}^{2}(\kappa,\lambda)\geq\lambda^{+}$* for any $\kappa,\lambda$ with $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$.* 6. (6)

$\mathrm{NT}_{\varphi}^{2}(\kappa,\lambda)\geq\lambda^{\kappa}$* for any $\kappa,\lambda$ such that $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$.*

Proof.

(1)$\Rightarrow$(6)  Assume $\varphi(x,y)$ has SOP2. Suppose that for infinite $\kappa,\lambda$, we have $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$. Hence $\kappa\leq\lambda$. We will show that $\mathrm{NT}^{2}_{\varphi}(\kappa,\lambda)\geq\lambda^{\kappa}$.

Since $\varphi$ has TP1 as in Fact 0.3(2), by compactness, there is a tree of formulas $\{\varphi(x,a_{\sigma})|\ \sigma\in\lambda^{<\kappa}\}$ witnessing TP1 w.r.t. $\lambda^{<\kappa}$  (i.e. for each $\beta\in\lambda^{\kappa}$, $q_{\beta}(x):=\{\varphi(x,a_{\beta\lceil i})\mid i<\kappa\}$ is consistent, while for any incomparable $\alpha,\gamma\in\lambda^{<\kappa}$, $\{\varphi(x,a_{\alpha}),\varphi(x,a_{\gamma})\}$ is inconsistent). Let $A$ be the set of parameters in the tree. We let ${\mathcal{F}}:=\{q_{\beta}(x)\mid\beta\in\lambda^{\kappa}\}$. Note that $|{\mathcal{F}}|=\lambda^{\kappa}>\lambda=\lambda^{<\kappa}=|A|.$

We want to show that ${\mathcal{F}}$ satisfies the condition (2) in Definition 1.1. Thus assume a set ${\mathcal{G}}=\{q_{\beta}\mid\beta\in\tau\}$ is given, where $\tau\subseteq\lambda^{\kappa}$ with $|\tau|=\lambda^{+}$. Now for each $\sigma\in\lambda^{<\kappa}$, we let ${\mathcal{G}}_{\sigma}:=\{p\in{\mathcal{G}}\mid\varphi(x,a_{\sigma})\in p\}$.

Claim. There are $\mu\in\lambda^{<\kappa}$ and $s_{0}<s_{1}\in\lambda$ such that $|{\mathcal{G}}_{\mu^{\smallfrown}\langle s_{0}\rangle}|=|{\mathcal{G}}_{\mu^{\smallfrown}\langle s_{1}\rangle}|=\lambda^{+}$: Suppose not. Then for each $\sigma\in\lambda^{<\kappa}$ there is at most one $s<\lambda$ such that $|{\mathcal{G}}_{\sigma^{\smallfrown}\langle s\rangle}|=\lambda^{+}$. Thus the only possibility is that there is $\delta\in\lambda^{\kappa}$ such that for each $i\in\kappa$, $|{\mathcal{G}}_{\delta\lceil i}|=\lambda^{+}$, while for each $j\in\lambda$ with $j\neq\delta(i)$, we have $|{\mathcal{G}}_{(\delta\lceil i)^{\smallfrown}\langle j\rangle}|\leq\lambda$. Since

[TABLE]

it follows that $|{\mathcal{G}}|\leq 1+\lambda\cdot\lambda\cdot\kappa=\lambda$, a contradiction. Hence the claim follows.

Now let $\tau_{0},\tau_{1}$ be the disjoint subsets of $\tau$ indexing the sets ${\mathcal{G}}_{\mu^{\smallfrown}\langle s_{0}\rangle}$ and ${\mathcal{G}}_{\mu^{\smallfrown}\langle s_{1}\rangle}$, respectively, so ${\mathcal{G}}_{\mu^{\smallfrown}\langle s_{j}\rangle}=\{p_{i}\in{\mathcal{G}}\mid i\in\tau_{j}\}$ ($j=0,1$). We now put for each $i\in\tau_{0}\cup\tau_{1}$, $p^{\prime}_{i}:=p_{i}\smallsetminus q_{\mu}$ where $q_{\mu}=\{\varphi(x,a_{\sigma})|\ \sigma\trianglelefteq\mu\}$. Hence $|p_{i}\smallsetminus p^{\prime}_{i}|<\kappa$. Moreover clearly each formula in $\bigcup_{i\in\tau_{1}}p^{\prime}_{i}$ is inconsistent with every formula in $\bigcup_{i\in\tau_{0}}p^{\prime}_{i}$. Therefore Definition 1.1(2) holds.

(6)$\Rightarrow$(5)$\Rightarrow$(2)$\Rightarrow$(4) and (6)$\Rightarrow$(3)$\Rightarrow$(2) Clear.

(4)$\Rightarrow$(1) Assume $\mathrm{NT}_{\varphi}^{2}(\kappa,\lambda)\geq\lambda^{+}$ for some infinite $\kappa$ and $\lambda$. Hence there is a family ${\mathcal{F}}=\{q_{i}\mid i<\lambda^{+}\}$ over a set $A$ with $|A|\leq\lambda$ satisfying the condition in Definition 1.1(2). We will produce an SOP2 tree for $\varphi$ from ${\mathcal{F}}$.

Claim. There exist a function $f:2^{<\omega}\to A$, a family $\{\,{\mathcal{G}}_{\sigma}\mid\sigma\in 2^{<\omega}\,\}$ of types, and a family $\,\{\,\tau_{\sigma}\subseteq\lambda^{+}\mid\sigma\in 2^{<\omega}\}\,$ such that for all $\sigma\in 2^{<\omega}$,

1. (i)

$|\tau_{\sigma}|=\lambda^{+}$; $\tau_{\sigma 0}$ and $\tau_{\sigma 1}$ are disjoint subsets of $\tau_{\sigma}$, 2. (ii)

${\mathcal{G}}_{\sigma}$ is of the form $\{p_{i}\mid p_{i}\subseteq q_{i},\ i\in\tau_{\sigma}\}$ (so $|{\mathcal{G}}_{\sigma}|=\lambda^{+}$) with $|q_{i}\smallsetminus p_{i}|<\kappa$, and for $j\in\{0,1\}$, ${\mathcal{G}}_{\sigma j}$ is of the form $\{p^{\prime}_{i}\mid p^{\prime}_{i}\subseteq p_{i}\in{\mathcal{G}}_{\sigma},\ i\in\tau_{\sigma j}\}$ with $|p_{i}\smallsetminus p^{\prime}_{i}|<\kappa$, 3. (iii)

for $a_{\sigma}:=f(\sigma)$ we have $\varphi(x,a_{\sigma})\in\bigcap{\mathcal{G}}_{\sigma}$, and 4. (iv)

each formula in $\bigcup{\mathcal{G}}_{\sigma 0}$ is inconsistent with every formula in $\bigcup{\mathcal{G}}_{\sigma 1}$.

Proof of Claim. We construct such a function and sets by induction on the length of $\sigma$. When $\sigma=\emptyset$, choose $\varphi(x,b_{i})$ from each $q_{i}\in{\mathcal{F}}$. Then since $|A|<\lambda^{+}$ and $\lambda^{+}$ is regular (or just by counting), there must be a subset $\tau_{\emptyset}\subseteq\lambda^{+}$ of size $\lambda^{+}$ such that $b_{i}$ are equal (say, to $a_{\emptyset}$) for all $i\in\tau_{\emptyset}$. Then set $f(\emptyset)=a_{\emptyset}$. Also, set ${\mathcal{G}}_{\emptyset}:=\{q_{i}\mid i\in\tau_{\emptyset}\}$, so $\varphi(x,a_{\emptyset})\in\bigcap{\mathcal{G}}_{\emptyset}$.

Assume now the induction hypothesis for $\sigma$. We will find sets and function values corresponding to $\sigma 0$ and $\sigma 1$. Write ${\mathcal{G}}_{\sigma}=\{p_{i}\mid i\in\tau_{\sigma}\}$. Since ${\mathcal{F}}$ satisfies Definition 1.1(2), there exist disjoint subsets $\tau^{\prime}_{\sigma j}\subseteq\tau_{\sigma}\,$ of size $\lambda^{+}$ ($j=0,1$) and a subset $p^{\prime}_{i}\subseteq q_{i}$ with $|q_{i}\smallsetminus p^{\prime}_{i}|<\kappa$ for each $i\in\tau^{\prime}_{\sigma 0}\cup\tau^{\prime}_{\sigma 1}$, such that every formula in $\bigcup_{i\in\tau^{\prime}_{\sigma 0}}p^{\prime}_{i}$ is inconsistent with each formula in $\bigcup_{i\in\tau^{\prime}_{\sigma 1}}p^{\prime}_{i}$. We now let $p^{\prime\prime}_{i}:=p_{i}\cap p^{\prime}_{i}$ for $i\in\bigcup_{j=0,1}\tau^{\prime}_{\sigma j}$, and let ${\mathcal{G}}^{\prime}_{\sigma j}:=\{p^{\prime\prime}_{i}\mid i\in\tau^{\prime}_{\sigma j}\}$. Then clearly $p^{\prime\prime}_{i}\subseteq p_{i}$, $|q_{i}\smallsetminus p^{\prime\prime}_{i}|<\kappa$, and $|p_{i}\smallsetminus p^{\prime\prime}_{i}|<\kappa$.

Now since again $|A|\leq\lambda$, for $j\in\{0,1\}$, there must be a set $\tau_{\sigma j}\subseteq\tau^{\prime}_{\sigma j}$ with $|\tau_{\sigma j}|=\lambda^{+}$ such that for some $d_{j}\in A$ (which we put $a_{\sigma j}=f(\sigma j)$), $\varphi(x,d_{j})\in\bigcap_{i\in\tau_{\sigma j}}p^{\prime\prime}_{i}$. Therefore if we let ${\mathcal{G}}_{\sigma j}:=\{p^{\prime\prime}_{i}\mid i\in\tau_{\sigma j}\}$, then $\tau_{\sigma j}$, $f(\sigma j)$ and ${\mathcal{G}}_{\sigma j}$, for $j=0,1$, satisfy all the required conditions for the induction step, and the proof for Claim is complete.

Now, using the properties described in Claim, we see that the tree $\{\varphi(x,a_{\sigma})\mid\sigma\in 2^{<\omega}\}$ witnesses SOP2. Indeed given any $\sigma,\beta,\gamma\in 2^{<\omega}$, the formula $\varphi(x,a_{\sigma^{\smallfrown}0^{\smallfrown}\beta})$ is inconsistent with $\varphi(x,a_{\sigma^{\smallfrown}1^{\smallfrown}\gamma})$. ∎

We now give type-counting criteria for SOP2, for a theory.

Theorem 1.3**.**

Let $\kappa,\lambda$ denote infinite cardinals. The following are equivalent.

1. (1)

$T$* has SOP2.* 2. (2)

For every regular $\kappa>|T|$, there is $\lambda\geq 2^{\kappa}$ such that $\mathrm{NT}^{2}(\kappa,\lambda)>\lambda$. 3. (3)

For some regular $\kappa>|T|$ and some $\lambda\geq 2^{\kappa}$, we have $\mathrm{NT}^{2}(\kappa,\lambda)>\lambda$. 4. (4)

For every $\kappa,\lambda$ with $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$, we have $\mathrm{NT}^{2}(\kappa,\lambda)\geq\lambda^{\kappa}$. 5. (5)

For every $\kappa,\lambda$ with $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$, we have $\mathrm{NT}^{2}(\kappa,\lambda)>\lambda$.

Proof.

(1)$\Rightarrow$(4)  The same proof of (1)$\Rightarrow$(6) for Theorem 1.2 shows this.

(2)$\Rightarrow$(3), (4)$\Rightarrow$(5)   Clear.

(3)$\Rightarrow$(1)   Assume (3) with the given $\kappa,\lambda$. Hence there is a family ${\mathcal{F}}$ of arbitrary types over $A$ with $|A|=\lambda$, satisfying the conditions (2) and (3) in Definition 1.1. There is no harm to assume that $|{\mathcal{F}}|=\lambda^{+}$ and we write ${\mathcal{F}}=\{q_{i}\mid i<\lambda^{+}\}.$ Since $|q_{i}|=\kappa$, we write $q_{i}=\{\varphi^{i}_{\alpha}(x,a^{i}_{\alpha})\mid\alpha<\kappa\},$ where $a^{i}_{\alpha}\in A$. Now since $|T|^{\kappa}=2^{\kappa}<\lambda^{+}$, there must be a subset $\tau$ of $\lambda^{+}$ with $|\tau|=\lambda^{+}$ such that the sequence $\langle\varphi^{i}_{\alpha}(x,y^{i}_{\alpha})\mid\alpha<\kappa\rangle$ stays the same, say $\langle\varphi_{\alpha}(x,y_{\alpha})\mid\alpha<\kappa\rangle$, for every $i\in\tau$. Moreover since $\kappa(>|T|)$ is regular, there must be a subset $\mu\subseteq\kappa$ of size $\kappa$ such that $\varphi_{\alpha}(x,y_{\alpha})$ stays the same, say $\varphi(x,y)$, for all $\alpha\in\mu$. Now we let ${\mathcal{F}}_{1}:=\{\{\varphi(x,a^{i}_{\alpha})\mid\alpha\in\mu\}\mid i\in\tau\}.$ Then it easily follows that ${\mathcal{F}}_{1}$ also satisfies Definition 1.1(1) and (2). Moreover each type in ${\mathcal{F}}_{1}$ is a positive $\varphi$-type. Therefore (1) follows by Theorem 1.2(4)$\Rightarrow$(1).

(5)$\Rightarrow$(2)   Assume (5). Now given regular $\kappa>|T|$, let $\lambda:=\beth_{\kappa}(\kappa)$. Then $\lambda^{<\kappa}=\lambda<\lambda^{\kappa}$. Hence by (5), we have $\mathrm{NT}^{2}(\kappa,\lambda)>\lambda$. ∎

2. Type-counting criteria for SOP1

As said in the beginning of Section 1, type-counting criteria for SOP2 are given in [12] as well. But for the first time, here we state and prove type-counting criteria for a formula to have SOP1.

Definition 2.1**.**

We say a formula $\varphi(x,y)$ has $\omega^{<\omega}$-SOP1 if there is a set $\{a_{\alpha}\mid\alpha\in\omega^{<\omega}\}$ of tuples such that

1. (1)

for each $\beta\in\omega^{\omega}$, $\{\varphi(x,a_{\beta\lceil n})\mid n\in\omega\}$ is consistent, and 2. (2)

for each $\beta\in\omega^{<\omega}$ and each pair $m<n\in\omega$, $\{\varphi(x,a_{\gamma}),\varphi(x,a_{\beta n})\}$ is inconsistent whenever $\beta m\unlhd\gamma$.

Fact 2.2**.**

A formula has SOP1 iff it has $\omega^{<\omega}$-SOP1.

Proof.

$(\Leftarrow)$ Clear.

$(\Rightarrow)$ Assume $\varphi(x,y)$ and $\{a_{\alpha}\mid\alpha\in 2^{<\omega}\}$ witness SOP1. Now for each $n>1$, define a $1-1$ map $f_{n}:n^{<\omega}\to 2^{<\omega}$ such that $f_{n}(\emptyset):=\emptyset$, and for $\alpha\in n^{<\omega}$ and $m<n$, $f_{n}(\alpha m):=f_{n}(\alpha)\overbrace{0\cdots 0}^{n-m-1}1$.

It follows that $A_{n}:=\{a_{f_{n}(\alpha)}\mid\alpha\in n^{<\omega}\}$ forms an $n^{<\omega}$-SOP1 tree for $\varphi$, and then compactness yields an $\omega^{<\omega}$-SOP1 tree for the formula. ∎

Definition 2.3**.**

Let $\varphi(x,y)\in{\mathcal{L}}$. For any two infinite cardinals $\kappa,\lambda$, we define $\mathrm{NT}_{\varphi}^{1}(\kappa,\lambda)$ as the supremum of cardinalities $|\mathcal{F}|$ of sets $\mathcal{F}$ of positive $\varphi$-types over some fixed set $A$ of cardinality $\lambda$ satisfying that

1. (1)

$|q(x)|=\kappa$ for every $q(x)\in\mathcal{F}$, and 2. (2)

given any subfamily ${\mathcal{G}}=\{q_{i}\mid i<\lambda^{+}\}$ of ${\mathcal{F}}$ and a family ${\mathcal{G}}^{\prime}=\{p_{i}\mid p_{i}\subseteq q_{i},\ i<\lambda^{+}\}$ where $|q_{i}\smallsetminus p_{i}|<\kappa$ for each $i<\lambda^{+}$, there are disjoint subsets $\tau_{0},\tau_{1}$ of $\lambda^{+}$ with $|\tau_{j}|=\lambda^{+}$ ($j=0,1$), and ${\mathcal{G}}^{\prime}_{j}=\{p^{\prime}_{i}\mid p^{\prime}_{i}\subseteq p_{i},\ i\in\tau_{j}\}$ with $|p_{i}\smallsetminus p^{\prime}_{i}|<\kappa$ for each $i\in\tau_{0}\cup\tau_{1}$, such that for every $p^{\prime}_{i}\in{\mathcal{G}}^{\prime}_{1}$ there is a formula in $p^{\prime}_{i}$ which is inconsistent with each formula in $\bigcup{\mathcal{G}}^{\prime}_{0}$.

Notice that if $|{\mathcal{F}}|\leq\lambda$ then the condition (2) is vacuous.

Theorem 2.4**.**

Assume $\varphi(x,y)$ is an ${\mathcal{L}}$-formula, and $\kappa,\lambda$ denote infinite cardinals. The following are equivalent.

1. (1)

$\varphi(x,y)$* has SOP1.* 2. (2)

$\mathrm{NT}_{\varphi}^{1}(\omega,\omega)\geq\omega_{1}$** 3. (3)

$\mathrm{NT}_{\varphi}^{1}(\omega,\omega)\geq 2^{\omega}.$** 4. (4)

$\mathrm{NT}_{\varphi}^{1}(\kappa,\lambda)\geq\lambda^{+}$* for some $\kappa,\lambda$.* 5. (5)

$\mathrm{NT}_{\varphi}^{1}(\kappa,\lambda)\geq\lambda^{+}$* for any $\lambda$ and any regular $\kappa$ with $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$.* 6. (6)

$\mathrm{NT}_{\varphi}^{1}(\kappa,\lambda)\geq\lambda^{\kappa}$* for any $\lambda$ and regular $\kappa$ such that $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$.*

Proof.

(1)$\Rightarrow$(6)  Assume $\varphi(x,y)$ has SOP1. Suppose that for regular $\kappa$, and infinite $\lambda$, we have $\lambda^{<\kappa}=\lambda$ and $\lambda^{\kappa}>\lambda$. We will show that $\mathrm{NT}^{1}_{\varphi}(\kappa,\lambda)\geq\lambda^{\kappa}$.

Since $\varphi$ has $\omega^{<\omega}$-SOP1 as in Fact 2.2, by compactness, there is a tree of formulas $\{\varphi(x,a_{\sigma})|\ \sigma\in\lambda^{<\kappa}\}$ witnessing SOP1 w.r.t. $\lambda^{<\kappa}$  (i.e. for each $\beta\in\lambda^{\kappa}$, $q_{\beta}(x):=\{\varphi(x,a_{\beta\lceil i})\mid i<\kappa\}$ is consistent, while for any $\alpha\in\lambda^{<\kappa}$ and $u<v\in\lambda$, $\{\varphi(x,a_{\gamma}),\varphi(x,a_{\alpha^{\smallfrown}\langle v\rangle})\}$ is inconsistent for any $\gamma\unrhd\alpha^{\smallfrown}\langle u\rangle$). Let $A$ be the set of parameters in the tree. We let ${\mathcal{F}}:=\{q_{\beta}(x)\mid\beta\in\lambda^{\kappa}\}$. Note that $|{\mathcal{F}}|=\lambda^{\kappa}>\lambda=\lambda^{<\kappa}=|A|.$

We want to show that ${\mathcal{F}}$ satisfies the condition (2) in Definition 2.3, Thus assume a set ${\mathcal{G}}=\{p_{\beta}\subseteq q_{\beta}\mid\beta\in\tau\}$ is given where $|q_{\beta}\smallsetminus p_{\beta}|<\kappa$ and $\tau\subseteq\lambda^{\kappa}$ with $|\tau|=\lambda^{+}$. Since $|q_{\beta}\smallsetminus p_{\beta}|<\kappa$ and $\kappa$ is regular, for each $\beta\in\tau$, there must exist an ordinal $i_{\beta}<\kappa$ such that $\{\varphi(x,a_{\beta\lceil i})\mid i_{\beta}\leq i<\kappa\}\subseteq p_{\beta}$. Note that $\lambda^{<\kappa}=\lambda$ implies $\kappa<\lambda^{+}$. Thus there exists a subset $\tau^{\prime\prime}\subseteq\tau$ of size $\lambda^{+}$ such that $i_{\beta}$ stays the same, say $i_{0}$ for every $\beta\in\tau^{\prime\prime}$. Once more, since $\lambda^{<\kappa}=\lambda$, for some subset $\tau^{\prime}\subseteq\tau^{\prime\prime}$ of size $\lambda^{+}$, $\beta\lceil i_{0}$ stays the same for every $\beta\in\tau^{\prime}$. Namely, there is $\sigma_{0}\in\lambda^{<\kappa}$ such that $\sigma_{0}=\beta\lceil i_{0}$ (and hence $a_{\sigma_{0}}=a_{\beta\lceil i_{0}}$) for all $\beta\in\tau^{\prime}$.

Now let ${\mathcal{G}}^{\prime}:=\{p_{\beta}\in{\mathcal{G}}\mid\beta\in\tau^{\prime}\}$, and for $\sigma(\unrhd\sigma_{0})\in\lambda^{<\kappa}$, we let ${\mathcal{G}}^{\prime}_{\sigma}:=\{p\in{\mathcal{G}}^{\prime}\mid\varphi(x,a_{\sigma})\in p\}$.

Claim. There are $\mu(\unrhd\sigma_{0})\in\lambda^{<\kappa}$ and $s_{0}<s_{1}\in\lambda$ such that $|{\mathcal{G}}^{\prime}_{\mu^{\smallfrown}\langle s_{0}\rangle}|=|{\mathcal{G}}^{\prime}_{\mu^{\smallfrown}\langle s_{1}\rangle}|=\lambda^{+}$: Suppose not. Thus for each $\sigma\unrhd\sigma_{0}\in\lambda^{<\kappa}$ there is at most one $s<\lambda$ such that $|{\mathcal{G}}^{\prime}_{\sigma^{\smallfrown}\langle s\rangle}|=\lambda^{+}$. Then it lead a contradiction by the similar cardinality computation in the proof of Claim in that of Theorem 1.2 (1)$\Rightarrow$(6). Hence the claim follows.

Now let $\tau_{0},\tau_{1}$ be the disjoint subsets of $\tau^{\prime}$ indexing the sets ${\mathcal{G}}^{\prime}_{\mu^{\smallfrown}\langle s_{0}\rangle}$ and ${\mathcal{G}}^{\prime}_{\mu^{\smallfrown}\langle s_{1}\rangle}$, respectively, so ${\mathcal{G}}^{\prime}_{\mu^{\smallfrown}\langle s_{j}\rangle}=\{p_{i}\in{\mathcal{G}}^{\prime}\mid i\in\tau_{j}\}$ ($j=0,1$). We now put for each $i\in\tau_{0}\cup\tau_{1}$, $p^{\prime}_{i}:=p_{i}\smallsetminus q_{\mu}$ where $q_{\mu}=\{\psi(x,a_{\sigma})|\ \sigma\trianglelefteq\mu\}$.

Notice that the formula $\varphi(x,a_{\mu s_{1}})\in\bigcap_{i\in\tau_{1}}p^{\prime}_{i}$ is inconsistent with any formula in $\bigcup_{i\in\tau_{0}}p^{\prime}_{i}$. Hence Definition 2.3(2) holds.

(6)$\Rightarrow$(5)$\Rightarrow$(2)$\Rightarrow$(4) and (6)$\Rightarrow$(3)$\Rightarrow$(2) Clear.

(4)$\Rightarrow$(1) Assume $\mathrm{NT}_{\varphi}^{1}(\kappa,\lambda)\geq\lambda^{+}$ for some infinite $\lambda$ and $\kappa$. Hence there is a family ${\mathcal{F}}=\{q_{i}\mid i<\lambda^{+}\}$ over a set $A$ with $|A|\leq\lambda$ satisfying the conditions Definition 2.3(1) and (2). We will produce an SOP1 tree for $\varphi$ from ${\mathcal{F}}$.

Claim. There exist a function $f:2^{<\omega}\to A$, a family $\{\,{\mathcal{G}}_{\sigma}\mid\sigma\in 2^{<\omega}\,\}$ of families of types, and a family $\,\{\,\tau_{\sigma}\subseteq\lambda^{+}\mid\sigma\in 2^{<\omega}\}\,$ such that, for all $\sigma\in 2^{<\omega}$,

1. (i)

$|\tau_{\sigma}|=\lambda^{+}$; $\tau_{\sigma 0}$ and $\tau_{\sigma 1}$ are disjoint subsets of $\tau_{\sigma}$, 2. (ii)

${\mathcal{G}}_{\sigma}$ is of the form $\{p_{i}\mid p_{i}\subseteq q_{i},\ i\in\tau_{\sigma}\}$ (so $|{\mathcal{G}}_{\sigma}|=\lambda^{+}$) with $|q_{i}\smallsetminus p_{i}|<\kappa$, and for $j\in\{0,1\}$, ${\mathcal{G}}_{\sigma j}$ is of the form $\{p^{\prime}_{i}\mid p^{\prime}_{i}\subseteq p_{i}\in{\mathcal{G}}_{\sigma},\ i\in\tau_{\sigma j}\}$ with $|p_{i}\smallsetminus p^{\prime}_{i}|<\kappa$, 3. (iii)

for $a_{\sigma}:=f(\sigma)$ we have $\varphi(x,a_{\sigma})\in\bigcap{\mathcal{G}}_{\sigma}$, and $\varphi(x,a_{\sigma 1})\in\bigcap{\mathcal{G}}_{\sigma 1}$ is inconsistent with every formula in $\bigcup{\mathcal{G}}_{\sigma 0}$.

Proof of Claim. We construct such a function and sets by induction on the length of $\sigma$. When $\sigma=\emptyset$, choose $\varphi(x,b_{i})$ from each $q_{i}\in{\mathcal{F}}$. Then since $|A|\leq\lambda$, there is a subset $\tau_{\emptyset}\subseteq\lambda^{+}$ of size $\lambda^{+}$ such that the $b_{i}$ are equal (say, to $a_{\emptyset}$) for all $i\in\tau_{\emptyset}$. Then set $f(\emptyset)=a_{\emptyset}$. Also, set ${\mathcal{G}}_{\emptyset}:=\{q_{i}\mid i\in\tau_{\emptyset}\}$, so $\varphi(x,a_{\emptyset})\in\bigcap{\mathcal{G}}_{\emptyset}$.

Assume now the induction hypothesis for $\sigma$. We will find sets and function values corresponding to $\sigma 0$ and $\sigma 1$. Write ${\mathcal{G}}_{\sigma}=\{p_{i}\mid i\in\tau_{\sigma}\}$. Since ${\mathcal{F}}$ satisfies Definition 2.3(2), there exist disjoint subsets $\tau^{\prime}_{\sigma j}\subseteq\tau_{\sigma}\,$ of size $\lambda^{+}$ ($j=0,1$) and a subset $p^{\prime}_{i}\subseteq p_{i}$ with $|p_{i}\smallsetminus p^{\prime}_{i}|<\kappa$ for each $i\in\tau^{\prime}_{\sigma 0}\cup\tau^{\prime}_{\sigma 1}$, such that for every $p^{\prime}_{i}\in{\mathcal{H}}_{1}$, there is a formula $\varphi(x,a^{\prime}_{i})\in p^{\prime}_{i}$ inconsistent with each formula in $\bigcup{\mathcal{H}}_{0}$, where ${\mathcal{H}}_{j}=\{p^{\prime}_{i}\mid i\in\tau^{\prime}_{\sigma j}\}$.

Now since again $|A|\leq\lambda$, there must be a set $\tau_{\sigma 1}\subseteq\tau^{\prime}_{\sigma 1}$ with $|\tau_{\sigma 1}|=\lambda^{+}$ such that $a^{\prime}_{i}$ are all equal for all $i\in\tau_{\sigma 1}$, which we put $f(\sigma 1)=a_{\sigma 1}$. Thus if we let ${\mathcal{G}}_{\sigma 1}:=\{p^{\prime}_{i}\mid i\in\tau_{\sigma 1}\}$, then $\varphi(x,a_{\sigma 1})\in\bigcap{\mathcal{G}}_{\sigma 1}$ is inconsistent with each formula in $\bigcup{\mathcal{H}}_{0}$. Similarly if we choose $\varphi(x,b^{\prime}_{i})\in q^{\prime}_{i}\in{\mathcal{H}}_{0}$, there must be a subset $\tau_{\sigma 0}\subseteq\tau^{\prime}_{\sigma 0}$ of size $\lambda^{+}$ such that $b^{\prime}_{i}$ stays the same for each $i\in\tau_{\sigma 0}$, which we let $f(\sigma 0)=a_{\sigma 0}$. Then let ${\mathcal{G}}_{\sigma 0}:=\{p^{\prime}_{i}\mid i\in\tau_{\sigma 0}\}$, so $\varphi(x,a_{\sigma 0})\in\bigcap{\mathcal{G}}_{\sigma 0}$. Therefore, $\tau_{\sigma j}$, $f(\sigma j)$ and ${\mathcal{G}}_{\sigma j}$, for $j=0,1$, satisfy all the required conditions for the induction step, and the proof for Claim is complete.

Now, using the properties described in Claim, we see that the tree $\{\varphi(x,a_{\sigma})\mid\sigma\in 2^{<\omega}\}$ witnesses SOP1. Indeed given any $\sigma\in 2^{<\omega}$, the formula $\varphi(x,a_{\sigma 1})$ is inconsistent with any $\varphi(x,a_{\gamma})$ where $\gamma\unrhd\sigma 0$. ∎

We finish this section by asking the following: Given a theory, are there criteria for SOP1 analogous to Theorem 1.3 for SOP2?

3. Kim-forking and TP2

We begin this section by recalling basic definitions.

Definition 3.1**.**

1. (1)

We say a formula $\varphi(x,a_{0})$ divides over a set $A$, if there is an $A$-indiscernible sequence $\langle a_{i}\mid i<\omega\rangle$ such that $\{\varphi(x,a_{i})\mid i<\omega\}$ is inconsistent. A formula forks over $A$ if the formula implies a finite disjunction of formulas, each of which divides over $A$. A type divides/forks over $A$ if the type implies a formula which divides/forks over $A$. We write $a\mathop{\smile\hskip-8.99994pt^{|}\ }_{A}B$ ($a\mathop{\smile\hskip-8.99994pt^{|}\ }^{d}_{A}B$) if $\operatorname{tp}(a/AB)$ *does not fork * (divide, resp.) over $A$. 2. (2)

An $A$-indiscernible sequence $\langle a_{i}\mid i<\omega\rangle$ is said to be a Morley sequence over $A$ if $a_{i}\mathop{\smile\hskip-8.99994pt^{|}\ }_{A}a_{<i}$ holds for each $i<\omega$. 3. (3)

We say a formula $\varphi(x,a_{0})$ Kim-divides over $A$ if $\{\varphi(x,a_{i})\mid i<\omega\}$ is inconsistent for some Morley sequence $\langle a_{i}\mid i<\omega\rangle$ over $A$. A formula Kim-forks over $A$ if the formula implies a finite disjunction of formulas, each of which Kim-divides over $A$. 4. (4)

A type Kim-divides/forks over $A$ if the type implies a formula which Kim-divides/forks over $A$. We write $c\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}_{A}B$ if $\operatorname{tp}(c/AB)$ *does not Kim-fork * over $A$. Hence $\mathop{\smile\hskip-8.99994pt^{|}\ }\Rightarrow\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}\mbox{ and }\mathop{\smile\hskip-8.99994pt^{|}\ }^{d}$.

Originally in [7], the notion of Kim-dividing is introduced over a model, using the notion of a Morley sequence in a global invariant extension of a type over the model. There it is shown that, over a model, that notion is equivalent to the one stated in Definition 3.1(3). Since in general even in a simple theory, there need not exist a global invariant extension of a type over a set, instead in [6] the above definition in (3) is coherently given as Kim-dividing over an arbitrary set.

As is well-known, in any simple $T$, $\mathop{\smile\hskip-8.99994pt^{|}\ }$ satisfies symmetry, full transitivity (that is: for any $d$ and $A\subseteq B\subseteq C$, $d\mathop{\smile\hskip-8.99994pt^{|}\ }_{A}B$ and $d\mathop{\smile\hskip-8.99994pt^{|}\ }_{B}C$ iff $d\mathop{\smile\hskip-8.99994pt^{|}\ }_{A}C$), extension, local character, finite character, and $3$-amalgamation of Lascar types. Moreover in such $T$, $\mathop{\smile\hskip-8.99994pt^{|}\ }=\mathop{\smile\hskip-8.99994pt^{|}\ }^{d}=\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}$ [10], and nonforking existence (that is: $d\mathop{\smile\hskip-8.99994pt^{|}\ }_{A}A$ for any $d$ and $A$) holds. As we will not deal with these facts, see [2] or [11] for more details. Further advances are discovered in [7],[8],[6],[4] recently. Namely, it is shown that in any NSOP1 $T$ having nonforking existence (as said any simple $T$, and all the known NSOP1 $T$ have this), the notions of Kim-forking and Kim-dividing coincide, and $\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}$ supplies a good independence notion since it satisfies all the aforementioned properties that hold of $\mathop{\smile\hskip-8.99994pt^{|}\ }$ in simple theories, except base monotonicity (so there can exist $d$ and $A\subseteq B\subseteq C$ such that $d\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}_{A}C$ but $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{B}C$ holds).

In this section we study TP2 in relation with Kim-forking. In particular we show that if $T$ has TP2 then there is a non-continuous Kim-forking chain of arbitrarily large length (Proposition 3.6), by which we prove that $T$ is supersimple iff there is no Kim-forking chain of length $\omega$ (Theorem 3.7). We also show that in any $T$ holding TP2, there is a type having arbitrarily large local weight with respect to $\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}$ (Proposition 3.8).

This section might be considered as an expository note, since all the results in this section are more or less straightforward consequences of known facts 3.3 and 3.5. In particular, the referee of this paper points out to us that Proposition 3.6 follows from a result in [3].

Recall that a sequence $\langle A_{i}\mid i<\kappa\rangle$ of sets is said to be continuous if for each limit $\delta<\kappa$, $A_{\delta}=\bigcup_{i<\delta}A_{i}$.

Fact 3.2**.**

[9]** The following are equivalent.

1. (1)

$T$* is NSOP1.* 2. (2)

There do not exist finite $d$ and a continuous increasing sequence $\langle M_{i}\mid i<|T|^{+}\rangle$ of $|T|$-sized models such that for each $i<|T|^{+}$, $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{M_{i}}M_{i+1}$.

Indeed the following is implicitly shown in [9] using Fact 0.4. We supply a proof for completeness.

Fact 3.3**.**

If $T$ has SOP1 then for each infinite cardinal $\kappa$, there exist a finite tuple $d$ and a continuous increasing sequence $\langle A_{\alpha}\mid\alpha<\kappa\rangle$ of sets such that for each $\alpha<\kappa$, $|A_{\alpha}|\leq|\alpha|\cdot\omega$ and $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{A_{\alpha}}A_{\alpha+1}$.

Proof.

Assume $T$ has SOP1. Given an infinite $\kappa$, by using compactness, there are a formula $\varphi(x,y)$ and an indiscernible sequence $\langle a_{i}b_{i}\mid i\in\mathbb{Z}\cdot\kappa\rangle$ satisfying Fact 0.4. Namely, $a_{i}\equiv_{(ab)_{<i}}b_{i}$ for all $i\in\mathbb{Z}\cdot\kappa$, $\{\varphi(x,a_{i})\mid i\in\mathbb{Z}\cdot\kappa\}$ is realized by say $d$, and $\{\varphi(x,b_{i})\mid i\in\mathbb{Z}\cdot\kappa\}$ is $2$-inconsistent  (*).

Now for $n<\omega$, let $A_{n}=\{a_{i}b_{i}\mid i\in\mathbb{Z}\cdot(n+1)\}$, and for $\omega\leq\alpha<\kappa$, let $A_{\alpha}=\{a_{i}b_{i}\mid i\in\mathbb{Z}\cdot\alpha\}$. Then clearly $\langle A_{\alpha}\mid\alpha\in\kappa\rangle$ is a continuous increasing sequence with $|A_{\alpha}|\leq\omega\cdot|\alpha|$. Moreover for each $b_{i}\in A_{\alpha+1}\setminus A_{\alpha}$, the countable sequence $I_{b_{i}}(\subset A_{\alpha+1}\setminus A_{\alpha})$ of successive $b_{j}$’s starting from $b_{i}$ is a finitely satisfiable indiscernible (so Morley) sequence in $\operatorname{tp}(b_{i}/A_{\alpha})$. Thus by (), $\varphi(x,b_{i})$ Kim-divides over $A_{\alpha}$. Then since $a_{i}\equiv_{A_{\alpha}}b_{i}$, again by () we have that $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{A_{\alpha}}a_{i}$. Note that $a_{i}\in A_{\alpha+1}\setminus A_{\alpha}$. Hence $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{A_{\alpha}}A_{\alpha+1}$ as wanted. ∎

Contrary to Fact 3.2(2), as in the following example, in NSOP1 $T$, there can exist a non-continuous increasing Kim-forking sequence of length $|T|^{+}$ of $\leq|T|$-sized sets, and arbitrary lengths continuous increasing Kim-forking sequences.

Example 3.4**.**

[4] Let $T$ be the theory of the random parametrized equivalence relations, i.e., the the Fraïssé limit of the class of finite models with two sorts $(P,E)$ and a ternary relation $\,\sim\,$ on $\,P\times P\times E\,$ such that, for each $e\in E$, $x\sim_{e}y$ forms an equivalence relation on $P$.

So in a model of $T$, there are two sorts $P$ and $E$ as described above. Let $d\in P$. Given a cardinal $\kappa$, choose distinct $e_{i}\in E$, and $d_{i}\in P$ ($i<\kappa)$ such that $d\sim_{e_{i}}d_{i}$, but $d_{j}\not\sim_{e_{k}}d_{i}$ for each $j<i$ and $k\leq i$. Let $D_{i}=((ed)_{<i})e_{i}$. Note that the sequence $\langle D_{i}\mid i<\kappa\rangle$ is increasing but not continuous (for example, $D_{<\omega}\subsetneq D_{\omega}$). Notice further that $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{D_{i}}d_{i}$, so $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{D_{i}}D_{i+1}$ for each $i<\kappa$.

Moreover, there is a continuous increasing Kim-forking sequence of length $\kappa$ of $\kappa$-sets. We work with the same chosen elements above. Let $C:=\{e_{i}\mid i<\kappa\}\subset E$, and let $C_{i}:=Cd_{<i}$. Clearly $\langle C_{i}\mid i<\kappa\rangle$ is a continuous increasing sequence of $\kappa$-sets. Now for each $i<\kappa$, it follows $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{C_{i}}d_{i}$, and hence $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{C_{i}}C_{i+1}$.

Now we can ask whether such phenomena happen in any non-simple NSOP1 $T$. We show that indeed in any theory with TP2, such sequences can be found. The following fact is well-known and a proof can be found for example in [13]. Recall that an array $\langle a_{ij}\mid i<\kappa,j<\lambda\rangle$ is said to be indiscernible111In some literature this notion is called strongly indiscernible over $A$ if for $L_{i}:=\langle a_{ij}\mid j<\lambda\rangle$, $\langle L_{i}\mid i<\kappa\rangle$ is $A$-indiscernible, and $A$-mutually indiscernible (i.e., each $L_{i}$ is indiscernible over $\bigcup\{L_{j}\mid j(\neq i)<\kappa\}A$).

Fact 3.5**.**

The following are equivalent.

1. (1)

$\varphi(x,y)$* has TP2.* 2. (2)

Let $\kappa$ be an infinite cardinal. There is an indiscernible array $\langle a_{ij}\mid i<\kappa,j<\omega+\omega\rangle$ such that

1. (a)

for each $i<\kappa$, $\{\varphi(x,a_{ij})\mid j<\omega+\omega\}$ is $2$-inconsistent, and 2. (b)

for any $f:\kappa\to\omega+\omega$, $\{\varphi(x,a_{if(i)})\mid i<\kappa\}$ is consistent.

Proposition 3.6**.**

Assume $T$ has TP2. Let $\kappa$ be an infinite cardinal.

1. (1)

There are a finite tuple $d$ and an increasing non-continuous sequence of sets $A_{i}$ $(i<\kappa)$ of size $|i|\cdot\omega(<\kappa)$ such that $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{A_{i}}A_{i+1}$ for each $i<\kappa$. In particular there is an increasing countable sequence of countable sets $B_{i}$ such that $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{B_{i}}B_{i+1}$ for each $i<\omega$. 2. (2)

There are a finite tuple $d$ and an increasing continuous sequence of sets $E_{i}$ $(i<\kappa)$ of size $\kappa$ such that $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{E_{i}}E_{i+1}$ for each $i<\kappa$.

Proof.

(1) Due to Fact 3.5 and compactness, there are a formula $\varphi(x,y)$ and an array $\langle a_{ij}\mid i<\kappa,\ j\in\omega+\omega^{*}\rangle$ where $\omega^{*}:=\{i^{*}\mid i\in\omega\}$ with the reversed order of $\omega$ (so for $i^{*},j^{*}\in\omega^{*}$, we have $n<i^{*}$ for all $n\in\omega$, and $j^{*}<i^{*}$ if $i<j$) such that

1. (a)

for each $i<\kappa$, $\{\varphi(x,a_{ij})\mid j\in\omega+\omega^{*}\}$ is $2$-inconsistent, 2. (b)

for any $f:\kappa\to\omega+\omega^{*}$, $\{\varphi(x,a_{if(i)})\mid i<\kappa\}$ is consistent, and 3. (c)

the array is mutually indiscernible, i.e., for any $i<\kappa$, $L_{i}:=\langle a_{ij}\mid j\in\omega+\omega^{*}\rangle$ is indiscernible over $\bigcup\{L_{j}\mid j(\neq i)<\kappa\}$.

For each $i\in\kappa$, we let $I_{i}:=\langle a_{ij}\mid j<\omega\rangle$, and let $J_{i}:=\langle a^{*}_{ij}\mid j<\omega\rangle$ where $a^{*}_{ij}=a_{ij^{*}}$ with $j^{*}\in\omega^{*}$, so as a set $L_{i}=I_{i}\cup J_{i}$.

Now due to (b), there is $d\models\{\varphi(x,a^{*}_{i0})\mid i<\kappa\}$. Put $A_{i}=\{I_{k}\mid k\leq i\}\cup\{a^{*}_{k0}\mid k<i\}$. Then $|A_{i}|=|i|\cdot\omega$. Now by (c), $J_{i}$ is finitely satisfiable, so Morley over $A_{i}$. Hence, by (a) we have

[TABLE]

for each $i<\kappa$. Notice that the sequence $\langle A_{i}\mid i<\kappa\rangle$ is not continuous, for example $A_{<\omega}=A_{\omega}\setminus I_{\omega}\subsetneq A_{\omega}$.

For the second statement of (1), simply put $B_{i}=A_{i}$ for $i<\omega$.

(2) We keep use the same $d$ in (1). Let $E:=I_{<\kappa}$, and for $i<\kappa$ let $E_{i}:=E\cup\{a^{*}_{k0}\mid k<i\}$. Now due to (c) again, for $i\in\kappa$, $J_{i}$ is Morley over $E_{i}$. Therefore we have

[TABLE]

as wanted. Note that clearly $\langle E_{i}\mid i<\kappa\rangle$ is a continuous increasing sequence with each $|E_{i}|=\kappa$. ∎

As said before Fact 3.2, the referee of this paper points out that Proposition 3.6 directly follows from the proof of Lemma 4.7 in [3] as well. Thus the above proof might be considered as the one describing the proposition as a straightforward consequence of Fact 3.5.

Now we recall that $T$ is supersimple if for any finite $a$, and a set $A$, there is a finite subset $A_{0}$ of $A$ such that $a\mathop{\smile\hskip-8.99994pt^{|}\ }_{A_{0}}A$. As is well-known $T$ is supersimple iff there does not exist a countably infinite forking chain (see for example, [11]). The following theorem says that the same holds with a countably infinite Kim-forking chain.

Theorem 3.7**.**

The following are equivalent.

1. (1)

$T$* is supersimple.* 2. (2)

There do not exist finite $d$ and an increasing sequence of sets $A_{i}$ ($i<\omega$) such that $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{A_{i}}A_{i+1}$ for each $i<\omega$. 3. (3)

There do not exist finite $d$ and an increasing sequence of countable sets $A_{i}$ ($i<\omega$) such that $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{A_{i}}A_{i+1}$ for each $i<\omega$.

Proof.

(1)$\Rightarrow$(2) is well-known as said before this theorem, and (2)$\Rightarrow$(3) is obvious.

(3)$\Rightarrow$(1)  We prove this contrapositively. Suppose $T$ is not supersimple. If $T$ is simple, then since $\mathop{\smile\hskip-8.99994pt^{|}\ }=\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}$, again it is well-known that there exist such a tuple and a sequence described in (3). If $T$ is NSOP1 but not simple, then $T$ has TP2 and Proposition 3.6 says there are such a tuple and a sequence. If $T$ has SOP1 then the existence of such a tuple and a sequence is guaranteed in Fact 3.3. ∎

As pointed out in [8], $T$ is NSOP1 iff there do not exist tuples $a_{i}$ ($i<\omega$), a model $M$, and an ${\mathcal{L}}$-formula $\varphi(x,y)$ such that for each $i<\omega$, $a_{i}\equiv_{M}a_{0}$, $a_{i}\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}_{M}a_{<i}$, $\varphi(x,a_{i})$ Kim-divides over $M$, and $\{\varphi(x,a_{i})\mid i<\omega\}$ is consistent. However only a slightly weaker condition always holds in any $T$ having TP2.

Proposition 3.8**.**

Assume $\varphi(x,y)$ has TP2. Then for each infinite $\kappa$, there are a set $A$ with $|A|\leq\kappa$, and finite tuples $d$, $c_{i}$ ($i<\kappa$) such that

1. (1)

$d\models\varphi(x,c_{i})$, 2. (2)

$\varphi(x,c_{i})$* Kim-divides over $A$ (so $d\mathop{\not\smile\hskip-8.99994pt^{|}\ }^{K}_{A}c_{i}$) witnessed by a Morley sequence $(c_{i}\in)J_{i}$ over $A$ with $J_{i}\equiv J_{0}$ (so $c_{i}\equiv c_{0}$), and* 3. (3)

$c_{i}\mathop{\smile\hskip-8.99994pt^{|}\ }_{A}\{c_{k}\mid k<\kappa,\ k\neq i\}$* (so $c_{i}\mathop{\smile\hskip-8.99994pt^{|}\ }^{K}_{A}\{c_{k}\mid k<\kappa,\ k\neq i\}$).*

Proof.

As in the proof of Proposition 3.6, there is an indiscernible array $\langle a_{ij}\mid i<\kappa,\ j\in\omega+\omega^{*}\rangle$ such that

1. (a)

for each $i<\kappa$, $\{\varphi(x,a_{ij})\mid j\in\omega+\omega^{*}\}$ is $2$-inconsistent, 2. (b)

for any $f:\kappa\to\omega+\omega^{*}$, $\{\varphi(x,a_{if(i)})\mid i<\kappa\}$ is consistent, and 3. (c)

for any $i<\kappa$, $L_{i}=\langle a_{ij}\mid j\in\omega+\omega^{*}\rangle$ is indiscernible over $\bigcup\{L_{j}\mid j<\kappa,\ j\neq i\}$.

Again for each $i\in\kappa$, let $I_{i}=\langle a_{ij}\mid j<\omega\}$, and $J_{i}=\langle a^{*}_{ij}\mid j<\omega\rangle$ where $a^{*}_{ij}=a_{ij^{*}}$ with $j^{*}\in\omega^{*}$. We further let $c_{i}:=a^{*}_{i0}$. Now by (b), there is $d\models\{\varphi(x,c_{i})\mid i<\kappa\}$.

We now put $A:=I_{<\kappa}$, so $|A|=\kappa$. Now due to (c), each $J_{i}$ is a Morley sequence over $A$ , and $\operatorname{tp}(c_{i}/A\{c_{k}\mid k<\kappa,\ k\neq i\})$ is finitely satisfiable in $A$. Hence (3) follows, and (2) follows as well due to (a). ∎