Tight cycles and regular slices in dense hypergraphs

TL;DR

This paper introduces regular slices in dense hypergraphs, demonstrating their utility in simplifying proofs and establishing new extremal results, including a hypergraph extension of the Erdős-Gallai Theorem and minimum codegree conditions for tight cycles.

Contribution

It develops the concept of regular slices in hypergraphs and applies them to prove new extremal hypergraph theorems and simplify existing proofs.

Findings

Proved a hypergraph extension of the Erdős-Gallai Theorem.

Determined asymptotic minimum codegree for tight cycles in k-partite hypergraphs.

Showed regular slices capture key structural properties of 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 $0<\alpha<1$.