Tight cycles and regular slices in dense hypergraphs

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 Erd\H{o}s-Gallai Theorem: for every $\delta>0$ every sufficiently large $k$-uniform hypergraph with at least $(\alpha+\delta)\binom{n}{k}$ edges contains a tight cycle of length $\alpha n$ for each $\alpha\in[0,1]$. 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 $\alpha kn$, for each $0<\alpha<1$.