Multicolour Turán problems

Abstract

A simple k-colouring of a multigraph G is a decomposition of the edge multiset as the sum of k simple graphs, called ‘colours’. A copy of some fixed graph H in G is called multicoloured if its edges all have distinct colours. Recall that the Turán number ex( n, H) of H is the maximum number of edges in a graph on n vertices not containing a copy of H. We consider a multicolour generalisation ex k ( n, H), defined as the maximum number of edges in a multigraph on n vertices, that has a simple k-colouring not containing a multicoloured copy of H. A natural construction of such a multigraph is k copies of a fixed extremal graph for H. We show that this is optimal for sufficiently large k= k( n), i.e., ex k ( n, H)= k·ex( n, H), and moreover only this construction achieves equality. For k⩽ e( H)−1 one can take k copies of the complete graph without creating a multicoloured copy of H, so this is trivially the best possible construction. Even for k⩾ e( H), we should consider a competing construction along these lines, namely e( H)−1 copies of the complete graph K n . When H= K r and n is large, the optimal construction is always one of these two, i.e., ex k(n,K r)= k· ex(n,K r) for k⩾(r 2−1)/2, ( r 2 −1)· n 2 for r 2 ⩽k<(r 2−1)/2. We prove a similar result for 3-colour-critical graphs. We also have some partial results for bipartite graphs. In particular, there are constants c< C so that for infinitely many values of n ex k(n,C 4)= k· ex(n,C 4) for k>C n , 3· n 2 for 4⩽k<c n .