A multiplicative inequality for vertex Folkman numbers

Abstract

Let G be a graph and a 1 , … , a r be positive integers. The symbol G → ( a 1 , … , a r ) denotes that in every r-coloring of the vertex set V ( G ) there exists a monochromatic a i -clique of color i for some i ∈ { 1 , … , r } . The vertex Folkman numbers F ( a 1 , … , a r ; q ) = min { | V ( G ) | : G → ( a 1 , … , a r ) and K q ⊈ G } are considered. Let a i , b i , c i , i ∈ { 1 , … , r } , s, t be positive integers and c i = a i b i , 1 ⩽ a i ⩽ s , 1 ⩽ b i ⩽ t . Then we prove that F ( c 1 , c 2 , … , c r ; st + 1 ) ⩽ F ( a 1 , a 2 , … , a r ; s + 1 ) F ( b 1 , b 2 , … , b r ; t + 1 ) .