New Lower Bounds for Two Multicolor Vertex Folkman Numbers

For a graph G, G → (α₁, α₂, ⋯, ar)v means that in every r-coloring of the vertices in G, there exists a monochromatic α_i-clique of color i for some i ∈ {1,2,⋯,r}. The vertex Folkman number is defined as Fν(α₁, α₂, ⋯ ,ar; k) = min{V(G) : G → (α₁, α₂, ⋯ , ar)^v and K_k ⊈ G}. In general, computing lower and upper bounds for vertex Folkman numbers is difficult. In this note, based on theoretical analysis and computation, we show that F_v(2,3,3;4) ≥ 19 and F_v(3,3,3;4) >; 24, and suggest a cvclic graph of order 91 which mav give an upper bound for F_v(3,3,3;4).