ON A CLASS OF VERTEX FOLKMAN GRAPHS (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2002)
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 $G→(a_1,...,a_r)$ means that in every $r$-colouring of the vertices of $G$ there exists a monochromatic $a_i$-clique of colour $i$, for some $i, 1 ≤ i ≤ r$. In this paper we consider the graphs $G→(a_1,...,a_r)$ (vertex Folkman graphs) with $cl(G) < m - 1$.