On the triangle vertex Folkman numbers

Abstract

For a graph G the symbol G→( 3,…,3 r ) means that in every r-colouring of the vertices of G there exists a monochromatic triangle. The triangle vertex Folkman numbers F r(3)= min{|V(G)|:G→( 3,…,3 r ) and cl( G)<2 r} are considered. We prove that F r (3)=2 r+7, r⩾3.