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ABSTRACT:We present a few results and a larger number of questions concerning partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously. We consider a variety of extremal problems; many of these also have algorithmic counterparts.

A tournament is a complete graph with its edges directed, and colouring a tournament means partitioning its vertex set into transitive subtournaments. For some tournaments H there exists c such that every tournament not containing H as a subtournament has chromatic number at most c (we call such a tournament H a hero); for instance, all tournaments with at most four vertices are heroes. In this paper we explicitly describe all heroes.

Gyárfás and Sumner independently conjectured that for every tree T and integer k there is an integer f(k, T) such that every graph G with χ(G) > f(k, t) contains either Kk or an induced copy of T. We prove a ‘topological’ version of the conjecture: for every tree T and integer k there is g(k,T) such that every graph G with χ(G) > g(k,t) contains either Kk or an induced copy of a subdivision of T. © 1997 John Wiley & Sons, Inc.