Multicoloured extremal problems

Many problems in extremal set theory can be formulated as finding the largest set system (or r-uniform set system) on a fixed ground set X that does not contain some forbidden configuration of sets. We shall consider multicoloured versions of such problems, defined as follows. Given a list of set systems, which we think of as colours, we call another set system multicoloured if for each of its sets we can choose one of the colours it belongs to in such a way that each set gets a different colour. Given an integer k and some forbidden configurations, the multicoloured extremal problem is to choose k colours with total size as large as possible subject to containing no multicoloured forbidden configuration. Let f be the number of sets in the smallest forbidden configuration. For kpf 1 we can take all colours to consist of all subsets of X (or all r-subsets in the uniform case), and this is trivially the best possible construction. Even for kXf 1; one possible construction is to take f 1 colours to consist of all subsets, and the other colours empty. Another construction is to take all k colours to be equal to a fixed family that is as large as possible subject to not containing a forbidden configuration. We shall consider a variety of problems in extremal set theory, for which we show that one of these two constructions is always optimal. This was shown for the multicoloured version of Sperner’s theorem by Daykin, Frankl, Greene and Hilton. We shall extend their result to some other Sperner problems, and also ARTICLE IN PRESS E-mail addresses: [email protected] [email protected] (B. Bollobás), [email protected] (P. Keevash), [email protected] (B. Sudakov). Research supported by NSF grant ITR 0225610 and DARPA grant F33615-01-C-1900. Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship. 0097-3165/$ see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcta.2004.05.003 prove multicoloured versions of the generalized Erdo+s–Ko–Rado theorem and the Sauer–Shelah theorem. r 2004 Elsevier Inc. All rights reserved.