Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 28, No 3, (2002), 219p-232p
Abstract:
Let a1 , . . . , ar, be positive integers, i=1 ... r, m = ∑(ai − 1) + 1 and
p = max{a1 , . . . , ar }. For a graph G the symbol G → (a1 , . . . , ar ) means
that in every r-coloring of the vertices of G there exists a monochromatic
ai -clique of color i for some i ∈ {1, . . . , r}. In this paper we consider the
vertex Folkman numbers
F (a1 , . . . , ar ; m − 1) = min |V (G)| : G → (a1 , . . . , ar ) and Km−1 ⊂ G}
We prove that F (a1 , . . . , ar ; m − 1) = m + 6, if p = 3 and m ≧ 6 (Theorem
3) and F (a1 , . . . , ar ; m − 1) = m + 7, if p = 4 and m ≧ 6 (Theorem 4).
