BOUNDS ON THE VERTEX FOLKMAN NUMBER F(4,4;5)

Abstract

For a graph $G$ the symbol $G\to (4,4)$ means that in every 2-coloring of the vertices of $G$ there exists a monochromatic $K_4$. For the vertex Folkman number $$F(4,4;5)=min\{|V(G)|:G\to (4,4)\text{and}{K}_5\not\subset G\}$$ we show that $16\leqq F(4,4;5)\leqq 35$.