On Some Generalized Vertex Folkman Numbers

For a graph $G$ and integers $a_i\ge 1$, the expression $G \rightarrow (a_1,\dots,a_r)^v$ means that for any $r$-coloring of the vertices of $G$ there exists a monochromatic $a_i$-clique in $G$ for some color $i \in \{1,\cdots,r\}$. The vertex Folkman numbers are defined as $F_v(a_1,\dots,a_r;H) = \min\{|V(G)| : G$ is $H$-free and $G \rightarrow (a_1,\dots,a_r)^v\}$, where $H$ is a graph. Such vertex Folkman numbers have been extensively studied for $H=K_s$ with $s>\max\{a_i\}_{1\le i \le r}$. If $a_i=a$ for all $i$, then we use notation $F_v(a^r;H)=F_v(a_1,\dots,a_r;H)$. Let $J_k$ be the complete graph $K_k$ missing one edge, i.e. $J_k=K_k-e$. In this work we focus on vertex Folkman numbers with $H=J_k$, in particular for $k=4$ and $a_i\le 3$. A result by Ne\v{s}et\v{r}il and R\"{o}dl from 1976 implies that $F_v(3^r;J_4)$ is well defined for any $r\ge 2$. We present a new and more direct proof of this fact. The simplest but already intriguing case is that of $F_v(3,3;J_4)$, for which we establish the upper bound of 135 by using the $J_4$-free process. We obtain the exact values and bounds for a few other small cases of $F_v(a_1,\dots,a_r;J_4)$ when $a_i \le 3$ for all $1 \le i \le r$, including $F_v(2,3;J_4)=14$, $F_v(2^4;J_4)=15$, and $22 \le F_v(2^5;J_4) \le 25$. Note that $F_v(2^r;J_4)$ is the smallest number of vertices in any $J_4$-free graph with chromatic number $r+1$. Most of the results were obtained with the help of computations, but some of the upper bound graphs we found are interesting by themselves.