A multiplicative inequality for vertex Folkman numbers

Let G   be a graph and a1,…,ara1,…,ar be positive integers. The symbol G→(a1,…,ar)G→(a1,…,ar) denotes that in every r  -coloring of the vertex set V(G)V(G) there exists a monochromatic aiai-clique of color i   for some i∈{1,…,r}i∈{1,…,r}. The vertex Folkman numbers F(a1,…,ar;q)=min{|V(G)|:G→(a1,…,ar)F(a1,…,ar;q)=min{|V(G)|:G→(a1,…,ar) and Kq⊈G}Kq⊈G} are considered. Let aiai, bibi, cici, i∈{1,…,r}i∈{1,…,r}, s, t   be positive integers and ci=aibici=aibi, 1⩽ai⩽s,1⩽ai⩽s,1⩽bi⩽t1⩽bi⩽t. Then we prove that F(c1,c2,…,cr;st+1)⩽F(a1,a2,…,ar;s+1)F(b1,b2,…,br;t+1).F(c1,c2,…,cr;st+1)⩽F(a1,a2,…,ar;s+1)F(b1,b2,…,br;t+1).