MAXIMUM NUMBER OF TRIANGLE-FREE EDGE COLOURINGS WITH FIVE AND SIX COLOURS

Let k >= 3 and r >= 2 be natural numbers. For a graph G, let F (G, k, r) denote the number of colourings of the edges of G with colours 1, ... , r such that, for every colour c 2 is an element of{1, ... , r}, the edges of colour c contain no complete graph on k vertices K-k. Let F (n, k, r) denote the maximum of F(G, k, r) over all graphs G on n vertices. The problem of determining F(n, k, r) was first proposed by Erdos and Rothschild in 1974, and has so far been solved only for r = 2, 3, and a small number of other cases. In this paper we consider the question for the cases k = 3 and r = 5 or r = 6. We almost exactly determine the value F (n, 3, 6) and approximately determine the value F (n, 3, 5) for large values of n. We also characterise all extremal graphs for r = 6 and prove a stability result for r = 5.