Let G be a graph and a1, ..., as be positive integers. The expression G v → (a1, ..., as) means that for every coloring of the vertices of G in s colors there exists i ∈ {1, ..., s} such that there is a monochromatic ai-clique of color i. The vertex Folkman numbers Fv(a1, ..., as; q) are defined by the equality: Fv(a1, ..., as; q) = min{|V (G)| : G v → (a1, ..., as) and Kq ⊆ G}.(ai − 1) + 1. It is easy to see that Fv(a1, ..., as; q) = m if q ≥ m+1. In [11] it is proved that Fv(a1, ..., as; m) = m+max{a1, ..., as}. We know all the numbers Fv(a1, ..., as; m − 1) when max{a1, ..., as} ≤ 5 and none of these numbers is known if max{a1, ..., as} ≥ 6. In this paper we compute all the numbers Fv(a1, ..., as; m−1) when max{a1, ..., as} = 6. We also prove that Fv(2, 2, 7; 8) = 20 and obtain new bounds on the numbers Fv(a1, ..., as; m − 1) when max{a1, ..., as} = 7.