An almost quadratic bound on vertex Folkman numbers

The vertex Folkman number F(r, n, m), n < m, is the smallest integer t such that there exists a Km-free graph of order t with the property that every r-coloring of its vertices yields a monochromatic copy of Kn. The problem of bounding the Folkman numbers has been studied by several authors. However, in the most restrictive case, when m = n + 1, no polynomial bound has been known for such numbers. In this paper we show that the vertex Folkman numbers F(r, n, n + 1) are bounded from above by O n2 log4 n . Furthermore, for any fixed r and any small ε > 0 we derive the linear upper bound when the cliques bigger than (2 + ε)n are forbidden. Key words: coloring of graphs, vertex Folkman numbers, generalized Ramsey theory

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Andrzej Dudek, Vojtech Rödl
• Keywords: Folkman Numbers, JCT 2010, Smallest Integer, Vertex Folkman Numbers