Abstract. We study properties of random subcomplexes of partitions returned by (a suitable form of) the Strong Hypergraph Regularity Lemma, which we call regular slices. We argue that these subcomplexes capture many important structural properties of the original hypergraph. Accordingly we advocate their use in extremal hypergraph theory, and explain how they can lead to considerable simplifications in existing proofs in this field. We also use them for establishing the following two new results.Firstly, we prove a hypergraph extension of the Erdős-Gallai Theorem: for every δ > 0 every sufficiently large k-uniform hypergraph with at least (α + δ) n k edges contains a tight cycle of length αn for each α ∈ [0, 1].Secondly, we find (asymptotically) the minimum codegree requirement for a k-uniform k-partite hypergraph, each of whose parts has n vertices, to contain a tight cycle of length αkn, for each 0 < α < 1.

Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [6, 23] the same result was proved for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs for any integer k ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform 'quasi-random' hypergraphs.

Chvátal, Rödl, Szemerédi and Trotter [V. Chvátal, V. Rödl, E. Szemerédi, W.T. Trotter Jr., The Ramsey number of a graph with a bounded maximum degree, J. Combin. Theory Ser. B 34 (1983) 239-243] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. We prove that the same holds for 3-uniform hypergraphs. The main new tool which we prove and use is an embedding lemma for 3-uniform hypergraphs of bounded maximum degree into suitable 3-uniform 'pseudo-random' hypergraphs.

In this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and consider the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We determine that the existence of a j-tuple-connected component containing Θ(n j ) j-sets in random k-uniform hypergraphs undergoes a phase transition and show that the threshold occurs at edge probability (k−j)! k j −1 n j−k . Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov which makes use of a depth-first search to reveal the edges of a random graph.Our main original contribution is a bounded degree lemma, which controls the structure of the component grown in the search process.

No abstract

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any k≥3 and 1≤j≤k−1.

No abstract

We consider the following definition of connectivity in k-uniform hypergraphs: Two j-sets are j-connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. We determine the threshold at which the random k-uniform hypergraph with edge probability p becomes j-connected with high probability. We also deduce a hitting time result for the random hypergraph process -the hypergraph becomes j-connected at exactly the moment when the last isolated j-set disappears. This generalises well-known results for graphs.