The vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1) = m + 9$, if $\max\{a_1, ..., a_s\} = 5$

Abstract

For a graph $G$ the expression $G \overset{v}{\rightarrow} (a_1, ..., a_s)$ means that for any $s$-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 $m = \sum_{i = 1}^{s}(a_i - 1) + 1$. With the help of computer we show that $F_v(2, 2, 5; 6) = 16$ and then we prove $$F_v(a_1, ..., a_s; m - 1) = m + 9,$$ if $\max\{a_1, ..., a_s\} = 5$. We also obtain the bounds $$m + 9 \leq F_v(a_1, ..., a_s; m - 1) \leq m + 10,$$ if $\max\{a_1, ..., a_s\} = 6$. Keywords: Folkman number, Ramsey number, clique number, independence number, chromatic number