Modified vertex Folkman numbers

Abstract

Let $a_1, ..., a_s$ be positive integers. For a graph $G$ the expression $$ G \overset{v}{\rightarrow} (a_1, ..., a_s) $$ means that for every coloring of the vertices of $G$ in $s$ colors ($s$-coloring) there exists $i \in \{1, ..., s\}$, such that there is a monochromatic $a_i$-clique of color $i$. If $m$ and $p$ are positive integers, then $$ G \overset{v}{\rightarrow} {m}\big\vert_{p} $$ means that for arbitrary positive integers $a_1, ..., a_s$ ($s$ is not fixed), such that $\sum_{i = 1}^{s}(a_i - 1) + 1 = m$ an $\max{\{a_1, ..., a_s\}} \leq p$ we have $G \overset{v}{\rightarrow} (a_1, ..., a_s)$. Let $$ \widetilde{\mathcal{H}}({m}\big\vert_{p}; q) = \{G : G \overset{v}{\rightarrow} {m}\big\vert_{p} \mbox{ and } \omega(G) < q\}. $$ The modified vertex Folkman numbers are defined by the equality $$ \widetilde{F}({m}\big\vert_{p}; q) = \min{\{|V(G)| : G \in \widetilde{\mathcal{H}}({m}\big\vert_{p}; q)\}}. $$ If $q \geq m$ these numbers are known and they are easy to compute. In the case $q = m - 1$ we know all of the numbers when $p \leq 5$. In this work we consider the next unknown case $p = 6$ and we prove with the help of a computer that $$ \widetilde{F}({m}\big\vert_{6}; m - 1) = m + 10. $$