ON A CLASS OF VERTEX FOLKMAN GRAPHS

Let $a_1,...,a_r$ be positive integers and $m=\sum\limits_{i=1}^{r}(a_i-1)+1$. For a graph $G$ the symbol $G\rightarrow(a_1,...,a_r)$ means that in every $r$-colouring of the vertices of $G$ there exists a monochromatic $a_i$-clique of colour $i$, for some $i, 1 \leq i \leq r$. In this paper we consider the graphs $G\rightarrow(a_1,...,a_r)$ (vertex Folkman graphs) with $\textrm{cl}(G) < m - 1$.