On the Vertex Folkman Numbers $F_v(2,\mathrm{.}\mathrm{.}\mathrm{.},2;q)$

For a graph $G$ the symbol $G\text{tov}(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_r)$ means that in every $r$-coloring of the vertices of $G$ for some $i\in \{1,\mathrm{.}\mathrm{.}\mathrm{.},r\}$ there exists a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers $$\text{FN}=\min \{\mathrm{\mid }V(G)\mathrm{\mid }:G\text{tov}(a_1,\mathrm{.}\mathrm{.}\mathrm{.},a_r)\text{and}{K}_q⊈G\}$$ are considered. In this paper we shall compute the Folkman numbers $F_v(\underset{r}{\underbrace{2,\mathrm{.}\mathrm{.}\mathrm{.},2}};r-k+1)$ when $k\le 12$ and $r$ is sufficiently large. We prove also new bounds for some vertex and edge Folkman numbers.