Finding Folkman Numbers via MAX CUT Problem

Abstract 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 K 4 -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 [J. Folkman, Graphs with monochromatic complete subgraphs in every edge coloring, SIAM J. Appl. Math., 18 (1970), 19–24, J. Nesetřil and V. Rodl, The Ramsey property for graphs with forbidden complete subgraphs, J. Comb. Theory Ser. B, 20, (1976), 243–249]. For almost twenty years the best known upper bound, given by Spencer, was f ( 2 , 3 , 4 ) 3 ⋅ 10 9 [J. Spencer, Three hundred million points suffice, J. Comb. Theory Ser. A, 49 (1988), 210–217. Also see erratum by M. Hovey in Vol. 50, p. 323]. Recently, the authors and Lu showed that f ( 2 , 3 , 4 ) 130000 [A. Dudek and V. Rodl, On the Folkman number f(2, 3, 4), submitted] and f ( 2 , 3 , 4 ) 10000 [L. Lu, Explicit construction of small Folkman graphs, submitted]. However, it is commonly believed that, in fact, f ( 2 , 3 , 4 ) 100 . All previous bounds are based on an idea of Goodman [A.W. Goodman, On sets of acquaintances and strangers at any party, Amer. Math. Mon., 66 (1959), 778–783]. 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 ) .