New Upper Bound for a Class of Vertex Folkman Numbers

Let a1, . . . , ar be positive integers, m = ∑r i=1(ai−1)+1 and p = max{a1, . . . , ar}. For a graph G the symbol G → {a1, . . . , ar} denotes that in every r-coloring of the vertices of G there exists a monochromatic ai-clique of color i for some i = 1, . . . , r. The vertex Folkman numbers F (a1, . . . , ar; m − 1) = min{|V (G)| : G → (a1 . . . ar) and Km−1 6⊆ G} are considered. We prove that F (a1, . . . , ar; m − 1) ≤ m + 3p, p ≥ 3. This inequality improves the bound for these numbers obtained by Luczak, Ruciński and Urbański (2001).