Tight cycles and regular slices in dense hypergraphs

Abstract

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.

Hypergraph extensions of the Erdős-Gallai Theorem

“…An extended abstract of our results already appeared in [9]. Since that several papers cited them in various context: [6,8,13], furthermore Allen, Böttcher, Cooley and Mycroft [1] have obtained upper and lower bounds for an analogue of Theorem 1.10 in the range k = αn for α constant, which are asymptotically almost equal when α is small.…”

“…Definition 1.9 ([10]). A tight path of length k in a r-uniform hypergraph is a collection of k + r − 1 vertices {v 1 …”

“…Let these edges be e 1 = {v 1 Thus we can assume that the greedy algorithm assigned an edge to each hyperedge. If H contains a path of length 4 on 5 different vertices then a hyperedges that were assigned to the edges to the path clearly form a path of length 4 in H, a contradiction again.…”

Regular slices for hypergraphs

“…In fact, the version of the Regular Slice Lemma stated in [1] is stronger in several useful ways. First, given several k-graphs G 1 , .…”

“…Finally, we can also insist that for any S ⊆ V (G) of size at most k − 1, the neighbourhood of S in G[J ] is an accurate representation of the neighbourhood of S in G (this final point is particularly useful for embedding spanning subgraphs). However, the form stated above is sufficient for the principal application given in [1], namely to prove a hypergraph analogue of the Erdős-Gallai theorem.…”