The vertex Folkman numbers $F_v(a_1, ..., a_s; m - 1) = m + 9$, if $\max\{a_1, ..., a_s\} = 5$

Abstract

For a graph G the expression G v → (a1, ..., as) means that for any scoloring of the vertices of G there exists i ∈ {1, ..., s} such that there is a monochromatic ai-clique of color i. The vertex Folkman numbers Fv(a1, ..., as; m − 1) = min{| V(G)| : G v → (a1, ..., as) and Km−1 ⊆ G}.are considered, where m = s i=1 (ai − 1) + 1. With the help of computer we show that Fv(2, 2, 5; 6) = 16 and then we prove Fv(a1, ..., as; m − 1) = m + 9, if max{a1, ..., as} = 5.We also obtain the bounds m + 9 ≤ Fv(a1, ..., as; m − 1) ≤ m + 10, if max {a1, ..., as} = 6.