On Some Generalized Vertex Folkman Numbers

Abstract

For a graph G and integers $$a_i\ge 1$$ , the expression $$G \rightarrow (a_1,\ldots ,a_r)^v$$ means that for any r-coloring of the vertices of G there exists a monochromatic $$a_i$$ -clique in G for some color $$i \in \{1,\ldots ,r\}$$ . The vertex Folkman numbers are defined as $$F_v(a_1,\ldots ,a_r;H) = \min \{|V(G)| : G$$ is H-free and $$G \rightarrow (a_1,\ldots ,a_r)^v\}$$ , 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 $$F_v(a^r;H)=F_v(a_1,\ldots ,a_r;H)$$ . 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 Nešetřil and Rödl 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 $$F_v(a_1,\ldots ,a_r;J_4)$$ 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.