Finding Folkman Numbers via MAX CUT Problem

In this note we report on our recent work, still in progress, regarding Folkman numbers. Let f(2, 3, 4) denote the smallest integer n such that there exists a K4– free graph of order n having that property that any 2–coloring of its edges yields at least one monochromatic triangle. It is well–known that such a number must exist [4,10]. For almost twenty years the best known upper bound, given by Spencer, was f(2, 3, 4) < 3 · 109 [13]. Recently, the authors and Lu showed that f(2, 3, 4) < 130 000 [2] and f(2, 3, 4) < 10 000 [9]. However, it is commonly believed that, in fact, f(2, 3, 4) < 100. All previous bounds are based on an idea of Goodman [6]. It seems that such methods will not yield substantial further improvement. In this note we will generalize this idea by giving a necessary and sufficient condition for a graph G to yield a monochromatic triangle for every edge coloring. In particular, for any graph G we construct a graph H such that G is Folkman if and only if the value of the maximum cut of H is less than twice the number of triangles in G. We believe this technique may be used to find a new upper bound on f(2, 3, 4).