ON THE VERTEX FOLKMAN NUMBERS Fv(2,...,2⏟R;R−1) and Fv(2,...,2⏟R;R−2)

Abstract

For a graph $G$ the symbol $G\stackrel{v}{\to }(a_1,...,a_r)$ means that in every $r$-coloring of the vertices of $G$ for some $i\in \{1,2,...,r\}$ there exists a monochromatic $a_i$-clique of color $i$. The vertex Folkman numbers $$F_v(\underset{r}{\underbrace{a_1,...,a_r}};q)=min\{|V(G)|:G\stackrel{v}{\to }(a_1,...,a_r)\text{ and }{K}_q⊈G\}$$ are considered. We prove that $$F_v(\underset{r}{\underbrace{2,...,2}};r-1)=r+7,r\ge 6\text{ and }{F}_v(\underset{r}{\underbrace{2,...,2}};r-2)=r+9,r\ge 8.$$