ABSTRACT:A hypergraph is b-simple if no two distinct edges share more than b vertices. Let m (r, t, g) denote the minimum number of edges in an r-uniform non-t-colorable hypergraph of girth at least g.Erdős and Lovász proved thatA result of Szabó improves the lower bound by a factor of r 2− for sufficiently large r. We improve the lower bound by another factor of r and extend the result to b-simple hypergraphs. We also get a new lower bound for hypergraphs with a given girth. Our results imply that for fixed b, t, and > 0 and sufficiently large r, every r-uniform b-simple hypergraph H with maximum edge-degree at most t r r 1− is t-colorable. Some results hold for list coloring, as well.

Given a set F of graphs, a graph G is F-free if G does not contain any member of F as an induced subgraph. We say that F is a degree-sequence-forcing set if, for each graph G in the class C of F-free graphs, every realization of the degree sequence of G is also in C. We give a complete characterization of the degree-sequence-forcing sets F when F has cardinality at most two.

A colouring of the vertices of a hypergraph H is called conflict-free if each edge e of H contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by χ CF (H). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph H with m edges, χ CF (H) is at most of the order of rm 1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

For a hypergraph G and a positive integer s, let χ (G, s) be the minimum value of l such that G is L-colorable from every list L withThis parameter was studied by Kratochvíl, Tuza, and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of χ (G, s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs. C 2013 Wiley Periodicals, Inc. J. Graph Theory 76: [129][130][131][132][133][134][135][136][137] 2014

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