On the 2-coloring diagonal vertex Folkman numbers with minimal possible clique number

Abstract

For a graph $G$ the symbol $G\stackrel{v}{\to }(p,p)$ means that in every 2-coloring of the vertices of $G$, there exists a monochromatic $p$-qlique. The vertex diagonal Folkman numbers $$F_v(p,p;p+1)=min\{|V(G)|:G\stackrel{v}{\to }(p,p)\text{and}{K}_{p+1}\not\subset G\}$$ are considered. We prove that $F_v(p,p;p+1)\le \frac{13}{12}p!,p\ge 4$.