Tur\'annical hypergraphs

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which deals with graphs G=([n],E) such that no member of the restriction set consisting of all r-tuples on [n] induces a copy of K_r. Firstly, we examine what happens when this restriction set is replaced just by all r-tuples touching a given m-element set. That is, we determine the maximal number of edges in an n-vertex such that no K_r hits a given vertex set. Secondly, we consider sparse random restriction sets. An r-uniform hypergraph R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if for any graph G on [n] with more edges than the Turan number ex(n,K_r) (respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G. We determine the thresholds for random r-uniform hypergraphs to be Turannical and to epsilon-Turannical. Thirdly, we transfer this result to sparse random graphs, using techniques recently developed by Schacht [Extremal results for random discrete structures] to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random graphs.Comment: 33 pages, minor improvements thanks to two referee

Matchings in 3-uniform hypergraphs

We determine the minimum vertex degree that ensures a perfect matching in a 3-uniform hypergraph. More precisely, suppose that H is a sufficiently large 3-uniform hypergraph whose order n is divisible by 3. If the minimum vertex degree of H is greater than \binom{n-1}{2}-\binom{2n/3}{2}, then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we show that H contains a matching of size d\le n/3 if its minimum vertex degree is greater than \binom{n-1}{2}-\binom{n-d}{2}, which is also best possible. This extends a result of Bollobas, Daykin and Erdos.Comment: 18 pages, 1 figure. To appear in JCT