Lower bounding the Folkman numbers $F_v(a_1, ..., a_s; m - 1)$

Abstract

For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for every $s$-coloring of the vertices of $G$ there exists $i \in \{1, ..., s\}$ such that there is a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers $$F_v(a_1, ..., a_s; m - 1) = \min\{\vert V(G)\vert : G \overset{v}{\rightarrow} (a_1, ..., a_s) \mbox{ and } K_{m - 1} \not\subseteq G\}.$$ are considered, where $m = \sum_{i = 1}^{s}(a_i - 1) + 1$. We know the exact values of all the numbers $F_v(a_1, ..., a_s; m - 1)$ when $\max\{a_1, ..., a_s\} \leq 6$ and also the number $F_v(2, 2, 7; 8) = 20$. In \cite{BN15a} we present a method for obtaining lower bounds on these numbers. With the help of this method and a new improved algorithm, in the special case when $\max\{a_1, ..., a_s\} = 7$ we prove that $F_v(a_1, ..., a_s; m - 1) \geq m + 11$ and this bound is exact for all $m$. The known upper bound for these numbers is $m + 12$. At the end of the paper we also prove the lower bounds $19 \leq F_v(2, 2, 2, 4; 5)$ and $29 \leq F_v(7, 7; 8)$.