Multicolour Turán problems

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 exk(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., exk(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 Kn. When H =Kr and n is large, the optimal construction is always one of these two, i.e., exk(n,Kr)= { k · ex(n,Kr) for k (r2 − 1)/2, ((r 2 )− 1) · (n2) for (r 2) k < (r2 − 1)/2. * Corresponding author. E-mail addresses: [email protected] (P. Keevash), [email protected] (M. Saks), [email protected] (B. Sudakov), [email protected] (J. Verstraëte). 1 Part of this research was done while the author was visiting Microsoft Research. Research supported in part by NSF grant 9988526. 2 Research supported in part by NSF grants DMS-0106589, CCR-9987845 and by the State of New Jersey. 0196-8858/$ – see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/j.aam.2003.08.005 P. Keevash et al. / Advances in Applied Mathematics 33 (2004) 238–262 239 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 exk(n,C4)= { k · ex(n,C4) for k > C √ n, 3 · (n2) for 4 k < c√n.  2003 Elsevier Inc. All rights reserved.