On the Nonexistence of Some Generalized Folkman Numbers

TL;DR

This paper investigates the existence and nonexistence of certain generalized Folkman numbers, providing proofs for some cases and posing open problems for others, advancing understanding in graph Ramsey theory.

Contribution

It establishes the well-definedness of specific edge and vertex Folkman numbers and proves the nonexistence of some, while proposing open problems for future research.

Findings

Proved certain generalized Folkman numbers are well defined for k ≥ 3.

Established nonexistence of specific Folkman numbers involving book graphs and other structures.

Formulated open problems on the existence of particular edge Folkman numbers.

Abstract

For an undirected simple graph $G$, we write $G \rightarrow (H_1, H_2)^v$ if and only if for every red-blue coloring of its vertices there exists a red $H_1$ or a blue $H_2$. The generalized vertex Folkman number $F_v(H_1, H_2; H)$ is defined as the smallest integer $n$ for which there exists an $H$-free graph $G$ of order $n$ such that $G \rightarrow (H_1, H_2)^v$. The generalized edge Folkman numbers $F_e(H_1, H_2; H)$ are defined similarly, when colorings of the edges are considered. We show that $F_e(K_{k+1},K_{k+1};K_{k+2}-e)$ and $F_v(K_k,K_k;K_{k+1}-e)$ are well defined for $k \geq 3$. We prove the nonexistence of $F_e(K_3,K_3;H)$ for some $H$, in particular for $H=B_3$, where $B_k$ is the book graph of $k$ triangular pages, and for $H=K_1+P_4$. We pose three problems on generalized Folkman numbers, including the existence question of edge Folkman numbers $F_e(K_3, K_3; B_4)$, $F_e(K_3, K_3; K_1+C_4)$ and $F_e(K_3, K_3; \overline{P_2 \cup P_3} )$. Our results lead to some general inequalities involving two-color and multicolor Folkman numbers.

Equations

$$F_e(3,3,k;s)\ge F_e(H,K_k;K_s).$$

Full text

We show that $F_{e}(K_{k+1},K_{k+1};K_{k+2}-e)$ and $F_{v}(K_{k},K_{k};K_{k+1}-e)$ are well defined for $k\geq 3$. We prove the nonexistence of $F_{e}(K_{3},K_{3};H)$ for some $H$, in particular for $H=B_{3}$, where $B_{k}$ is the book graph of $k$ triangular pages, and for $H=K_{1}+P_{4}$. We pose three problems on generalized Folkman numbers, including the existence question of edge Folkman numbers $F_{e}(K_{3},K_{3};B_{4})$, $F_{e}(K_{3},K_{3};K_{1}+C_{4})$ and $F_{e}(K_{3},K_{3};\overline{P_{2}\cup P_{3}})$. Our results lead to some general inequalities involving two-color and multicolor Folkman numbers.

Keywords: Folkman number, Ramsey number

AMS classification subjects: 05C55, 05C35

1 Introduction

Let $G$ be a finite undirected graph that contains no loops or multiple edges. Denote by $V(G)$ the set of its vertices and $E(G)$ the set of its edges. For vertex-disjoint graphs $G$ and $H$, the join graph $G+H$ has the set of vertices $V(G)\cup V(H)$ and edges $E(G)\cup E(H)\cup\{\{(u,v\}\ |\ u\in V(G),v\in V(H)\}$. For a set of vertices $S\subset V(G)$, $G[S]$ is the graph induced by $S$ in $G$, and $G-u$ is the graph obtained from $G$ by removing vertex $u\in V(G)$ together with all the edges adjacent to $u$.

The complete graph of order $n$ is denoted by $K_{n}$, and a cycle of length $n$ by $C_{n}$. The book graph $B_{k}$ is defined as $K_{1}+K_{1,k}$, and the complete graph $K_{n}$ with one missing edge will be denoted by $J_{n}$. The clique number of $G$ will be denoted by $cl(G)$, and the chromatic number of $G$ by $\chi(G)$. An $(s,t)$-graph is a graph that does not contain $K_{s}$ neither any independent sets of $t$ vertices. The set $\{1,\cdots,n\}$ will be denoted by $[n]$.

For graph $G$, we write $G\rightarrow(H_{1},H_{2})^{v}$ if and only if for every red-blue coloring $\chi$ of the vertices $V(G)$ there exists a red subgraph $H_{1}$ or a blue subgraph $H_{2}$ in $\chi$. The generalized vertex Folkman number $F_{v}(H_{1},H_{2};H)$ is defined as the smallest integer $n$ for which there exists an $H$-free graph $G$ of order $n$ such that $G\rightarrow(H_{1},H_{2})^{v}$. The set of all $H$-free graphs satisfying the latter vertex arrowing will be denoted by $\mathcal{F}_{v}(H_{1},H_{2};H)$.

The generalized edge Folkman numbers $F_{e}(H_{1},H_{2};H)$ are defined similarly, when colorings of the edges are considered. We write $G\rightarrow(H_{1},H_{2})^{e}$ if and only if for every red-blue coloring $\chi$ of the edges $E(G)$ there exists a red subgraph $H_{1}$ or a blue subgraph $H_{2}$ in $\chi$. The generalized edge Folkman number $F_{e}(H_{1},H_{2};H)$ is defined as the smallest integer $n$ for which there exists an $H$-free graph $G$ of order $n$ such that $G\rightarrow(H_{1},H_{2})^{e}$. The set of all $H$-free graphs satisfying the latter edge arrowing will be denoted by $\mathcal{F}_{e}(H_{1},H_{2};H)$.

The cases when $H_{1}$, $H_{2}$ and $H$ are complete graphs have been studied by many authors, for two and more colors, in particular in [1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 15, 17]. Often, if the graphs $H_{i}$ and $H$ are complete, we will simply write the order of the graph, say, as in $F_{e}(s,t;k)$ instead of $F_{e}(K_{s},K_{t};K_{k})$. In this paper we focus on two colors, but we will also make some comments related to more colors, such as in commonly studied multicolor vertex Folkman numbers $F_{v}(a_{1},a_{2},\cdots,a_{r};s)$ and edge Folkman numbers $F_{e}(a_{1},a_{2},\cdots a_{r};s)$, where $a_{i}$’s are the orders of the arrowed complete graphs while coloring $K_{s}$-free graphs. We note that the classical Ramsey number $R(a_{1},\cdots,a_{r})$ can be defined as the smallest integer $n$ such that $K_{n}\rightarrow(a_{1},\cdots,a_{r})^{e}$. In the diagonal case $a_{1}=\cdots=a_{r}=a$ we may use a more compact notation $F_{v}^{r}(a;s)=F_{v}(a_{1},\cdots,a_{r};s)$ and $\mathcal{F}_{v}^{r}(a;s)=\mathcal{F}_{v}(a_{1},\cdots,a_{r};s)$, similarly $F_{e}^{r}(a;s)=F_{e}(a_{1},\cdots,a_{r};s)$ and $\mathcal{F}_{e}^{r}(a;s)=\mathcal{F}_{e}(a_{1},\cdots,a_{r};s)$, as well as for arrowing general graphs, such as in $F_{e}^{r}(G;H)$.

In 1970, Folkman [5] proved that for any integer $s>\max\{a_{1},\cdots,a_{r}\}$, both sets $\mathcal{F}_{v}(a_{1},\cdots,a_{r};s)$ and $\mathcal{F}_{e}(a_{1},a_{2};s)$ are nonempty, and thus the corresponding Folkman numbers are well defined. In 1976, Nešetřil and Rödl [12] generalized this result to the multicolor edge cases, namely they proved that the sets $\mathcal{F}_{e}(a_{1},\cdots,a_{r};s)$ are also nonempty, for arbitrary $r\geq 2$ and $s>\max\{a_{1},\cdots,a_{r}\}$. An interesting upper bound on $F_{v}^{r}(a;s)$ was obtained by Dudek and Rödl [2] in 2010, as in the following theorem.

Theorem 1**.**

[2]* For any positive integer $r$ there exists a constant $C=C(r)$ such that for every $s\geq 2$ it holds that $F_{v}^{r}(s;s+1)\leq Cs^{2}\log{{}^{4}}s$.*

The above determines that both vertex and edge Folkman numbers exist when the arrowed and avoided graphs are complete, for $s>\max\{a_{1},\cdots,a_{r}\}$. By simple monotonicity, this easily extends to some cases (say, when the arrowed graphs $H_{i}$ have at most $a_{i}$ vertices), but apparently it poses interesting existence questions in other cases. Only some special parameters are discussed in the literature, such as the bound $F_{e}(K_{4}-e,K_{4}-e;K_{4})\leq 30193$ obtained by Lu [8] in 2008. In this paper we focus on some general situations, in particular when the avoided graph $H$ is not the complete graph $K_{s}$, but $H_{1}$ and $H_{2}$ are complete, and often $H_{1}=H_{2}=K_{3}$.

The nonexistence of a Folkman number with some parameters is equivalent to the emptiness of the corresponding set of Folkman graphs. For example, the Folkman number $F_{e}(K_{3},K_{3};K_{1}+P_{4})$ does not exist if and only if $\mathcal{F}_{e}(K_{3},K_{3};K_{1}+P_{4})=\emptyset$, which in fact we prove to be true in Theorem 8, Section 4.

The summary of contents of the remainder of this paper is as follows: Vertex and edge arrowing by $(K_{s}-e)$-free graphs and related existence questions are discussed in Section 2, similarly for graphs involving book graphs in Section 3. Other cases involving wheels and paths are analyzed in Section 4. Finally, in Section 5 some results for more than two colors are presented.

2 Ramsey arrowing by $(K_{n}-e)$-free graphs

Recall from the introduction that $J_{k}=K_{k}-e$. One can easily see that $F_{v}(2,2;3)=F_{v}(K_{2},K_{2};K_{3})=5$, which can be equivalently stated as that the smallest number of vertices in any triangle-free graph $G$ with $\chi(G)>2$ is equal to 5. However, it is also easy to observe that $F_{v}(K_{2},K_{2};J_{3})$ does not exist, since every $J_{3}$-free graph is bipartite. Similarly, we see that $F_{e}(K_{3},K_{3};J_{4})$ does not exist, since in any $J_{4}$-free graph no two triangles can share an edge, and thus the edges of every triangle can be independently red-blue colored. These observations lead to our first theorem.

Theorem 2**.**

For $k\geq 3$, if the edge Folkman number $F_{e}(K_{k+1},K_{k+1};J_{k+2})$ exists, then the vertex Folkman number $F_{v}(K_{k},K_{k};J_{k+1})$ exists too.

Proof.

Suppose that $F_{e}(K_{k+1},K_{k+1};J_{k+2})$ exists, it is equal to $n$, and let $G$ be any graph of order $n$ in $\mathcal{F}_{e}(K_{k+1},K_{k+1};J_{k+2})$. For any vertex $u\in V(G)$ we must have $G-u\not\rightarrow(K_{k+1},K_{k+1})^{e}$. Fix any vertex $u\in V(G)$, and let $H$ be the graph induced in $G$ by the neighbors of $u$, $H=G[N(u)]$. Clearly, $H$ is a $J_{k+1}$-free graph.

For contradiction, assume that $F_{v}(K_{k},K_{k};J_{k+1})$ does not exist. This implies that $H\not\rightarrow(k,k)^{v}$, and hence there exists a partition of $N(u)$ into $U_{1}\cup U_{2}$ such that both $G[U_{1}]$ and $G[U_{2}]$ are $K_{k}$-free. Next, observe that any red-blue edge coloring witnessing $G-u\not\rightarrow(K_{k+1},K_{k+1})^{e}$ can be extended to whole $E(G)$, without creating any monochromatic $K_{k+1}$, by coloring the edges $\{\{u,v\}\in E(G)\;|\;v\in U_{1}\}$ red and coloring the edges $\{\{u,v\}\in E(G)\;|\;v\in U_{2}\}$ blue. This contradicts that $G\in\mathcal{F}_{e}(K_{k+1},K_{k+1};J_{k+2})$, and completes the proof. ∎

Graph $H$ is called a Ramsey graph for $K_{n}$ if $H\rightarrow(K_{n},K_{n})^{e}$. In 1981, Nešetřil and Rödl [13] proved the following theorem.

Theorem 3**.**

[13]* Let $n\geq 3$ be a fixed positive integer. Then there exists a Ramsey graph $H$ for $K_{n}$ such that any two subgraphs $K$, $K^{\prime}$ of $H$ isomorphic to $K_{n}$ intersect in at most two points.*

Corollary 4**.**

For every integer $k\geq 3$,

(a)* the edge Folkman number $F_{e}(K_{k+1},K_{k+1};J_{k+2})$ exists, and*

(b)* the vertex Folkman number $F_{v}(K_{k},K_{k};J_{k+1})$ exists.*

Proof.

Graph $H$ in Theorem 3 does not contain $J_{n+1}$ for $n\geq 4$, thus if $n=k+1$ then the set $\mathcal{F}_{e}(K_{k+1},K_{k+1};J_{k+2})$ is nonempty, and hence part (a) of the corollary follows. Theorem 2 and part (a) imply part (b). ∎

We can easily see that for integers $s$ and $t$, if $k>s\geq t\geq 2$, then $F_{v}(K_{s},K_{t};J_{k+1})$ exists, and by monotonicity $F_{v}(K_{s},K_{t};J_{k+1})\leq F_{v}(s,t;k)$. The upper bound for $F_{e}(K_{k+1},K_{k+1};J_{k+2})$ which can be obtained using the proof of Theorem 3 is large, and likely it is much larger than the exact value. Similarly, the implied upper bound for $F_{v}(K_{k},K_{k};J_{k+1})$ is likely much larger than the exact value. It would be interesting to obtain better upper bounds for these numbers directly without using Theorem 3, for example by a method similar to one used in the proof of Theorem 1 in [2]. We note that a straightforward reasoning similar to a method used in [6] leads to an inequality $F_{v}(K_{s_{1}s_{2}},K_{t_{1}t_{2}};$ $J_{k_{1}k_{2}+1})\leq F_{v}(s_{1},t_{1};k_{1}+1)F_{v}(K_{s_{2}},K_{t_{2}};J_{k_{2}+1}),$ for $2\leq s_{1}\leq t_{1}\leq k_{1}$ and $3\leq s_{2}\leq t_{2}\leq k_{2}$. This makes us anticipate that $F_{v}(K_{k},K_{k};J_{k+1})$ grows slowly with $k$, and possibly can be bounded by $cF_{v}(k,k;k+1)$ for some constant $c>0$.

The best known concrete lower and upper bounds on various Ramsey numbers of the form $R(J_{s},K_{t})$ are collected in [14]; for example, we know that $30\leq R(J_{5},K_{5})\leq 33$. In that case, any 29-vertex witness graph to Ramsey lower bound seems to be a good candidate for the vertex Folkman number case of arrowing $(3,4)^{v}$. This would give an interesting bound $F_{v}(K_{3},K_{4};J_{5})\leq 29$ (unfortunately, we were not successful in finding any such graph so far). Still we think that, in general, further exploration of witnesses to lower bounds for Ramsey numbers as graphs showing upper bounds for (vertex or edge) Folkman numbers is worth an effort.

3 Arrowing triangles by $B_{k}$-free graphs

Recall that the book graph $B_{k}$ was defined as $B_{k}=K_{1}+K_{1,k}$, hence it has $k+2$ vertices and consists of $k$ triangles sharing one common edge. In particular, $B_{1}=K_{3}$, $B_{2}=J_{4}$ and $B_{3}=K_{5}\setminus K_{3}$. Thus, the first book-specific case (different from $K_{k}$ and $J_{k}$) is that for the book graph $B_{3}$ considered in the next theorem.

Theorem 5**.**

There exists a $B_{3}$-free and $K_{4}$-free graph $G$ of order $19$ such that $G\rightarrow(3,3)^{v}$. Thus we have $F_{v}(K_{3},K_{3};B_{3})\leq 19$.

Note. For the upper bound in the second part of the theorem it is not required that the graph $G$ is $K_{4}$-free. In any case, we consider the bound in Theorem 5 quite strong. Finding the actual value of $F_{v}(K_{3},K_{3};B_{3})$ can be difficult, and it is open whether the best construction must contain $K_{4}$.

Proof.

We will construct the required graph $G$ on the vertex set $V(G)=\bigcup_{i=0}^{3}V_{i}\cup\{u\}$, where $G[V_{0}]=K_{3}$, and the subgraphs induced by $V_{i}$ are isomorphic to $C_{5}$, for $i\in\{1,2,3\}$. Let $V_{0}=\{v_{1},v_{2},v_{3}\}$. The other edges of $G$ are all possible edges between $u$ and $\bigcup_{i=1}^{3}V_{i}$, and all possible edges between $v_{i}$ and the vertices in $V_{i}$, for each $i\in\{1,2,3\}$. Thus $G$ has 19 vertices and 48 edges.

It is easy to see that the graph $G$ is both $K_{4}$-free and $B_{3}$-free. With little more effort, one can show that every red-blue coloring of $V(G)$ contains a monochromatic triangle. Without loss of generality we can assume that $u$ is red and at least one of the vertices in $V_{0}$, say $v_{1}$, is blue. However, in order to avoid a red triangle on $u$ and two vertices in $V_{1}$, $G[V_{1}]$ must contain a blue $K_{2}$. But the latter together with $v_{1}$ would form a blue triangle. Hence we have that $G\rightarrow(3,3)^{v}$. Finally, the same graph $G$ is a witness of the upper bound. ∎

Since $B_{2}=J_{4}$, and using the observation from the beginning of Section 2, we see that $F_{e}(K_{3},K_{3};B_{2})$ does not exist. Now we will consider the existence of $F_{e}(K_{3},K_{3};B_{k})$ for $k\geq 3$, starting with the case of $B_{3}$.

Theorem 6**.**

The edge Folkman number $F_{e}(K_{3},K_{3};B_{3})$ does not exist.

Proof.

Suppose that $F_{e}(K_{3},K_{3};B_{3})$ exists, it is equal to $n$, and let $G$ be any graph of order $n$ in $\mathcal{F}_{e}(K_{3},K_{3};B_{3})$. For any vertex $u\in V(G)$ we must have $G-u\not\rightarrow(K_{3},K_{3})^{e}$. Fix any vertex $u\in V(G)$, and let $H$ be the graph induced in $G$ by the neighbors of $u$, $H=G[N(u)]$. Since $G$ is $B_{3}$-free, $H$ does not contain $K_{1,3}$, or equivalently has maximum degree at most 2. Therefore any connected component of $H$ is bipartite or it is an odd cycle.

We will show that any red-blue coloring $\chi$ of the edges of $G-u$, such that $\chi$ is without monochromatic triangles, can be extended to $G$ without creating any monochromatic triangles. This will contradict the definition of $G$ and thus it will complete the proof.

For the edges $\{u,v\}$, where $v$ is in a bipartite component of $H$, we assign the color red or blue according to which part of the bipartition $v$ belongs to. For vertices $v$ on odd cycles in $H$, we proceed as follows. Let $U$ be the vertex set of some odd cycle in $H$. We can partition $U$ into $U_{1}\cup U_{2}$ so that $H[U_{1}]$ has exactly one edge, say $e$, and $U_{2}$ is an independent set in $H$. If $\chi(e)$ is red (blue), then we color the edges in $\{\{u,v\}\;|\;v\in U_{1}\}$ blue (red), and the edges in $\{\{u,v\}\;|\;v\in U_{2}\}$ red (blue). ∎

We were not able to answer the question whether $F_{e}(K_{3},K_{3};B_{4})$ exists, and hence we leave it as an open problem for the readers. Note that for every $k\geq 5$, the edge Folkman number $F_{e}(K_{3},K_{3};B_{k})$ exists, and it is equal to 6, because the complete graph $K_{6}$ is $B_{k}$-free and $K_{6}\rightarrow(K_{3},K_{3})^{e}$.

Problem 3.1**.**

Does the edge Folkman number $F_{e}(K_{3},K_{3};B_{4})$ exist?

In Theorem 5 we constructed a $K_{4}$-free and $B_{3}$-free graph $G$ vertex arrowing $(3,3)^{v}$. We think that it is an interesting challenge to solve the following graph existence problem for $K_{4}$-free and book-free graphs edge arrowing $(3,3)^{e}$.

Problem 3.2**.**

For which $k\geq 4$ there exists a $K_{4}$-free and $B_{k}$-free graph $G$ such that $G\rightarrow(3,3)^{e}$?

The answer seems not easy even just for $k=4$. Note that a YES solution to Problem 3.1 does not provide an answer to Problem 3.2 with $k=4$, while a NO answer to Problem 3.1 implies a NO answer to Problem 3.2 for $k=4$. For Problem 3.2, we know that the answer is NO for $k=3$ by Theorem 6 (hence we ask only about cases for $k\geq 4$), and clearly a YES answer for any $k$ would imply YES answers for all $t>k$.

One of the most wanted Folkman numbers is $F_{e}(3,3;4)=F_{e}(K_{3},K_{3};K_{4})$, for which the currently best known bounds are $20\leq F_{e}(3,3;4)$ [1] and $F_{e}(3,3;4)$ $\leq 786$ [7]. The value of $F_{e}(3,3;4)$ can be equivalently defined as the smallest number of vertices in any $K_{4}$-free graph which is not a union of two triangle-free graphs. An overview of what is known about this problem was presented in [16]. In particular, it was conjectured by Exoo that a special cubic residues $(4,12)$-graph $G_{127}$ on the vertex set $\mathcal{Z}_{127}$ is a witness to a much improved upper bound $F_{e}(3,3;4)\leq 127$, and likely its subgraphs may even give $F_{e}(3,3;4)\leq 94$ (see [16]). The graph $G_{127}$ is $K_{4}$-free, has independence number 11, is $B_{12}$-free, but it contains a large number of subgraphs isomorphic to $B_{11}$. The Exoo’s conjecture can be stated as $G_{127}\rightarrow(3,3)^{e}$. If true, then it would give a YES answer in Problem 3.2 for all $k\geq 12$, leaving open the cases for $4\leq k\leq 11$. Recall that by Theorem 6 the answer for $k=3$ is NO.

4 More on arrowing triangles

In this section we study the existence of $F_{e}(K_{3},K_{3};H)$ for connected graphs $H$. First, we observe that, since graph avoidance is monotonic with respect to subgraphs, if a graph $H$ is connected and $cl(H)\geq 4$, then there exist $H$-free graphs edge arrowing $(3,3)^{e}$, i.e. $F_{e}(K_{3},K_{3};H)$ exists, and obviously $F_{e}(K_{3},K_{3};H)\leq F_{e}(3,3;4)$. For 5 vertices, there are 4 such graphs, namely $\widehat{K}_{4,i}$ for $i\in[4]$, where $\widehat{K}_{n,s}$ is the graph obtained by connecting a new vertex $v$ to $s$ vertices of a $K_{n}$. Clearly, the numbers $F_{e}(K_{3},K_{3};\widehat{K}_{4,i})$ exist for $i\in[4]$, and $F_{e}(K_{3},K_{3};\widehat{K}_{4,i+1})\leq F_{e}(K_{3},K_{3};\widehat{K}_{4,i})$ for $1\leq i\leq 3$. In particular, note that $\widehat{K}_{4,3}=J_{5}$, $\widehat{K}_{4,4}=K_{5}$, and we have the easy bounds $15=F_{e}(3,3;5)\leq F_{e}(K_{3},K_{3};J_{5})\leq F_{e}(3,3;4)\leq 786$, using only what is known about $F_{e}(3,3;k)$ [16]. For $\widehat{K}_{4,i}$-free graphs, $i=1,2$, we have the following lemma.

Lemma 7**.**

$F_{e}(K_{3},K_{3};\widehat{K}_{4,2})=F_{e}(K_{3},K_{3};\widehat{K}_{4,1})=F_{e}(3,3;4).$**

Proof.

By the monotonicity of $F_{e}(K_{3},K_{3};\widehat{K}_{4,i})$ mentioned above, it is sufficient to prove that $F_{e}(K_{3},K_{3};\widehat{K}_{4,2})\geq F_{e}(3,3;4)$. We will show that for any graph $G\in\mathcal{F}_{e}(K_{3},K_{3};\widehat{K}_{4,2})$ there exists a subgraph $G^{\prime}\in\mathcal{F}_{e}(3,3;4)$ of $G$, which will complete the proof. Define graph $G^{\prime}$ on the same set of vertices as $G$, with the set of edges $E(G^{\prime})=E(G)\setminus\{e\;|\;e\in K_{4}\subset G\}$. Obviously, $G^{\prime}$ is $K_{4}$-free. Since $G$ is $\widehat{K}_{4,2}$-free, we can see that every triangle in $G$ which is not a triangle in $G^{\prime}$ has its three vertices in the same $K_{4}$ of $G$. Thus, any red-blue edge coloring of $E(G^{\prime})$ without monochromatic triangles can be extended to whole $E(G)$ by independently red-blue coloring the edges of each $K_{4}$. This contradicts that $G\in\mathcal{F}_{e}(K_{3},K_{3};\widehat{K}_{4,2})$. Thus, no such coloring of $E(G^{\prime})$ exists, and hence $G^{\prime}\in\mathcal{F}_{e}(3,3;4)$. ∎

In the remainder of this section, we will consider only connected graphs $H$ with $K_{3}$ but without $K_{4}$. There are three such graphs on 4 vertices, namely $J_{4}$ and its subgraphs, and hence as commented in Section 2, $F_{e}(K_{3},K_{3};H)$ does not exist in these cases. In the following, we focus attention on connected graphs $H$ of order 5 with $cl(H)=3$, and leave the study of such graphs with more than 5 vertices for future work. The next theorem claims the nonexistence of $F_{e}(K_{3},K_{3};H)$ for a special 5-vertex graph $H=K_{1}+P_{4}$.

Theorem 8**.**

The edge Folkman number $F_{e}(K_{3},K_{3};K_{1}+P_{4})$ does not exist.

Proof.

The proof is very similar to that of Theorem 6. Suppose contrary, that $F_{e}(K_{3},K_{3};K_{1}+P_{4})$ exists, it is equal to $n$, and let $G$ be any graph of order $n$ in $\mathcal{F}_{e}(K_{3},K_{3};K_{1}+P_{4})$. For any vertex $u\in V(G)$ we must have $G-u\not\rightarrow(K_{3},K_{3})^{e}$. Fix any vertex $u\in V(G)$, and let $H$ be the graph induced in $G$ by the neighbors of $u$, $H=G[N(u)]$. Since $G$ is $(K_{1}+P_{4})$-free, $H$ does not contain $P_{4}$. Therefore any connected component of $H$ is bipartite or isomorphic to $K_{3}$. Now, the same steps as in the proof of Theorem 6 lead to a contradiction. ∎

We now state a theorem summarizing the existence of $F_{e}(K_{3},K_{3};H)$ for all connected graphs $H$ on 5 vertices with $cl(H)=3$. Only two cases remain open, namely those for the wheel graph $W_{5}$ and the complement of $P_{2}\cup P_{3}$. These cases should be studied more, and we expect that new insights can be important for better understanding of which graphs edge arrow $(3,3)^{e}$.

Theorem 9**.**

Let $H$ be any connected $K_{4}$-free graph on $5$-vertices containing $K_{3}$. Then the edge Folkman number $F_{e}(K_{3},K_{3};H)$ does not exist, except for two possible cases for $H$, namely $W_{5}$ and $\overline{P_{2}\cup P_{3}}$.

Proof.

There are 11 nonisomorphic $K_{4}$-free connected graphs on 5 vertices containing $K_{3}$. By Theorem 8, $F_{e}(K_{3},K_{3};K_{1}+P_{4})$ does not exist. The graph $K_{1}+P_{4}$ contains as a subgraph 7 further such graphs $H$ (including the bowtie graph $K_{1}+2K_{2}$, $K_{1,4}+e$, and the so-called bull graph), for which by monotonicity $F_{e}(K_{3},K_{3};H)$ does not exist either. This leaves three cases: $B_{3}$, $W_{5}$ and $\overline{P_{2}\cup P_{3}}$. The first case was eliminated by Theorem 6, while the other two are as the stated exceptions. ∎

Problem 4.1**.**

Prove or disprove the existence

(a)* of the edge Folkman number $F_{e}(K_{3},K_{3};\overline{P_{2}\cup P_{3}})$, and*

(b)* of the edge Folkman number $F_{e}(K_{3},K_{3};K_{1}+C_{4})$.*

Note that $W_{5}=K_{1}+C_{4}$ is a subgraph of $J_{5}=K_{5}-e$. Hence, if $F_{e}(K_{3},K_{3};W_{5})$ exists, then we have $F_{e}(K_{3},K_{3};J_{5})\leq F_{e}(K_{3},K_{3};W_{5})$. The analogous statement holds for the complement of $P_{2}\cup P_{3}$. On the other hand, the latter is a subgraph of $W_{5}$, hence there are only three possible combined YES/NO answers to the existence questions (a) and (b) in Problem 4.1, namely NO/NO, YES/YES and NO/YES.

A natural direction to generalize considerations of this section is to analyze which small graphs on at least $6$ vertices necessarily are subgraphs of every $K_{4}$-free graph edge arrowing $(3,3)^{e}$. The simplest candidate for such a graph is $B_{4}=K_{6}\setminus K_{4}$, as stated in Problem 3.1. One could also proceed by making a catalog of small subgraphs in known witnesses of existence of $F_{e}(3,3;4)$, in particular for the graph $G_{786}$, which currently is the smallest known such graph [7]. This, and even only some conditional answers to our problems, may lead to better bounds on $F_{e}(3,3;4)$.

5 Some cases of multicolor Ramsey arrowing

Since we know that $F_{e}(K_{3},K_{3};J_{4})$ does not exist, if a 3-color edge arrowing $G\rightarrow(K_{3},K_{3},K_{k})^{e}$ holds, then we must have $G\rightarrow(J_{4},K_{k})^{e}$. This easily generalizes to $F_{e}(K_{3},K_{3},K_{k};K_{s})\geq F_{e}(J_{4},K_{k};K_{s})$ for $s>k\geq 3$, and in particular it gives $F_{e}(3,3,3;4)\geq F_{e}(J_{4},K_{3};K_{4})$. We note that $F_{e}(3,3,3;4)$ exists, its value is unknown, it is likely quite large, and probably still much harder to obtain than the notoriously difficult case of $F_{e}(3,3;4)$. Clearly, the same reasoning holds for any graph $H$ instead of $J_{4}$ for which $F_{e}(K_{3},K_{3};H)$ does not exist, including $B_{3}$, $K_{1}+P_{4}$ or other graphs discussed in the previous section. This leads to the following corollary.

Corollary 10**.**

If $H$ is any graph for which $F_{e}(K_{3},K_{3};H)$ does not exist, then for $s>k\geq 3$ we have

[TABLE]

Proof.

As in the comments above, we observe that any $n$-vertex graph $G$ witnessing the upper bound $F_{e}(3,3,k;s)\leq n$ must also satisfy $G\rightarrow(H,K_{k})^{e}$. Thus we have $F_{e}(H,K_{k};K_{s})\leq n$. ∎

It would be interesting to construct a $K_{4}$-free graph $G$ such that $G\rightarrow(K_{3},J_{4})^{e}$ but $G\not\rightarrow(3,3,3)^{e}$. This might be quite hard since it is difficult to construct any $K_{4}$-free graph that arrows $(K_{3},J_{4})^{e}$, and it would be another challenge to show that it does not arrow $(3,3,3)^{e}$. Similarly, obtaining any nontrivial lower bound for the difference $F_{e}(3,3,3;4)-F_{e}(K_{3},J_{4};K_{4})$ seems difficult.

On the other hand, there exists an interesting example of a $K_{4}$-free graph $G$ on 30193 vertices, constructed by Lu [8], such that $G\rightarrow(J_{4},J_{4})^{e}$ (thus also $G\rightarrow(K_{3},J_{4})^{e}$). It is possible that for this graph we have $G\rightarrow(3,3,3)^{e}$, however we do not know how to prove or disprove the latter. Also, note that by an argument as in the proof of Corollary 10 we have $F_{e}^{4}(3;4)=F_{e}(3,3,3,3;4)\geq F_{e}(J_{4},J_{4};K_{4})$.

Finally, we establish a new link between some two-color edge Folkman numbers and multicolor vertex Folkman numbers. They generalize a result obtained in [17].

Lemma 11**.**

For $k\geq s\geq 2$ and graphs $G$ and $H$, if $G$ is $H$-free, $H\subset K_{k+1}$, and $G\rightarrow(K_{s},K_{k})^{e}$, then for every vertex $u\in V(G)$ and $s-1$ colors we have $G-u\rightarrow(K_{k},\cdots,K_{k})^{v}$.

Proof.

For a contradiction, suppose that for some graphs $G$ and $H$ as specified in the lemma, and for some vertex $u\in V(G)$, there exists a partition $V(G-u)=\bigcup_{i=1}^{s-1}V_{i}$, such that the graphs $G[V_{i}]$ are $K_{k}$-free, for every $i\in[s-1]$.

Now, we color red or blue all the edges in $E(G)$ as follows. All edges in each $G[V_{i}]$, for $i\in[s-1]$, are colored blue. The edges in $G[N(u)]$ are also blue. The edges between $u$ and $N(u)$ are red, and all other edges in $E(G)$, which are necessarily between different parts $V_{i}$, are also colored red. Note that any red clique may have at most one vertex in each of the parts $V_{i}$, and that there are no red triangles passing through vertex $u$. Thus, this coloring has no red $K_{s}$. No nontrivial blue clique contains vertex $u$, and none of $G[V_{i}]$ contains blue $K_{k}$, hence any potential blue $K_{k}$ on vertices $S$ must intersect different parts $V_{i}$. However, if such $S$ exists, and because of how the coloring was defined, the set of vertices $S\cup\{u\}$ would form a $K_{k+1}$, contrary to the assumption that $G$ is $H$-free. ∎

Corollary 12**.**

For $2\leq s\leq k$ and graph $H\subset K_{k+1}$, if $F_{e}(K_{s},K_{k};H)$ exists, then $F_{v}^{s-1}(K_{k};H)$ also exists and $F_{e}(K_{s},K_{k};H)\geq F_{v}^{s-1}(K_{k};H)+1$.

Proof.

Consider any graph $G$, such that $G\in\mathcal{F}_{e}(K_{s},K_{k};H)$, of the least possible order $F_{e}(K_{s},K_{k};H)$. Then by Lemma 11, the graph $G-u$ is in the set $\mathcal{F}_{v}^{s-1}(K_{k};H)$ and it has one vertex less than $G$. This proves the inequality. ∎

The proofs of our last lemma and corollary use a method similar to one applied in the proof of $F_{e}(3,k;k+1)>F_{v}(k,k;k+1)$ in [17]. The latter is a special case of Corollary 12 with $s=3$ and $H=K_{k+1}$. Another interesting instantiation of Corollary 12 is for $H=J_{k+1}$, for which the existence question of corresponding Folkman numbers was discussed in Section 2.