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

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 \[ F(4,4;5)=\min\{|V(G)| : G\to(4,4)\ \mbox {and}\ K_5\not\subset G\} \] we show that $16\leqq F(4,4;5)\leqq35$.