On the vertex Folkman numbers $F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;m-1)$ when $\max \{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s\}=6$ or $7$

Let $G$ be a graph and $a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s$ be positive integers. Then $G \overset{v}{\rightarrow} (a_1, …, a_s)$ means that for every coloring of the vertices of $G$ in $s$ colors there exists $i\in \{1,\mathrm{.}\mathrm{.}\mathrm{.},s\}$, such that there is a monochromatic $a_i$-clique of color $i$. The vertex Folkman number $F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;q)$ is defined by the equality: $$F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;q)=\min \{\mathrm{\mid }V(G)\mathrm{\mid }:G(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s)and{K}_q\subseteq ̸G\}\mathrm{.}$$ Let $m={\sum }_{i=1}^s(a_i-1)+1$. It is easy to see that $F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;q)=m$ if $q\ge m+1$. In [11] it is proved that $F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;m)=m+\max \{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s\}$. We know all the numbers $F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;m-1)$ when $\max \{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s\}\le 5$ and none of these numbers is known if $\max \{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s\}\ge 6$. In this paper we compute the numbers $F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;m-1)$ when $\max{a_1, …, a_s} = 6$.