Maximal antiramsey graphs and the strong chromatic number

Abstract

AbstractA typical problem arising in Ramsey graph theory is the following. For given graphs G and L, how few colors can be used to color the edges of G in order that no monochromatic subgraph isomorphic to L is formed? In this paper we investigate the opposite extreme. That is, we will require that in any subgraph of G isomorphic to L, all its edges have different colors. We call such a subgraph a totally multicolored copy of L. Of particular interest to us will be the determination of Xs(n, e, L), defined to be the minimum number of colors needed to edge‐color some graph G(n, ϵ) with n vertices and e edges so that all copies of L in it are totally multicolored.It turns out that some of these questions are surprisingly deep, and are intimately related, for example, to the well‐studied (but little understood) functions rk(n), defined to be the size of the largest subset of {1, 2,…, n} containing no k‐term arithmetic progression, and g(n; k, l), defined to be the maximum number of triples which can be formed from {1, 2,…, n} so that no two triples share a common pair, and no k elements of {1, 2,…, n} span l triples.