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

For a graph $G$ the expression $G(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s)$ means that for every $s$-coloring of the vertices of $G$ there exists $i \in {1, …, s}$suchthatthereisamonochromatic$a_i$-cliqueofcolor$i$. The vertex Folkman numbers $$F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;m-1)=\min \{\mathrm{\mid }V(G)\mathrm{\mid }:G(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s)and{K}_{m-1}\subseteq ̸G\}\mathrm{.}$$ are considered, where $m={\sum }_{i=1}^s(a_i-1)+1$. We know the exact values of all the numbers $F_v(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s;m-1)$ when $\max \{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s\}\le 6$ 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 $\max \{a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_s\}=7$ we prove that $F_v(a_1, …, a_s; m - 1) \geq m + 11$andthisboundisexactforall$m$. 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$29 \leq F_v(7, 7; 8)$.