Let a1, . . . , ar be positive integers, m = âˆ‘r i=1(aiâˆ’1)+1 and p = max{a1, . . . , ar}. For a graph G the symbol G â†’ {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 âˆ’ 1) = min{|V (G)| : G â†’ (a1 . . . ar) andâ€¦ (More)

Tort in [4] drew attention to the problem of the minimum number of vertices of the 5-chromatic graphs, not containing 4-cliques. Hanson et al. in [1] solve this problem by proving that the minimum number of vertices equals 11. But perhaps they were not aware that in paper of Nenov [3] given in the references under N 3 by them, the problem has already beenâ€¦ (More)

Let G be a graph and V (G) be the vertex set of G. Let a1 ,. . . , ar be positive integers, m = âˆ‘ r i=1 (aiâˆ’1)+1 and p = max{a1, . . . , ar}. The symbol G â†’ {a1, . . . , ar} denotes that in every r-coloring of V (G) there exists a monochromatic ai-clique of color i for some i = 1, . . . , r. The vertex Folkman numbers F (a1, . . . , ar ; m âˆ’ 1) = min{|Vâ€¦ (More)

In [6] we computed the edge Folkman number F (3, 4; 8) = 16. There we used and announced without proof that in any blue-red coloring of the edges of the graph K1 +C5 +C5 + C5 there is either a blue 3-clique or red 4-clique. In this paper we give a detailed proof of this fact.