Cycles of given lengths in hypergraphs

TL;DR

This paper introduces a new method for analyzing cycle lengths in hypergraphs, proving a conjecture about the existence of Berge cycles of consecutive lengths in hypergraphs with certain degree conditions.

Contribution

The paper develops a novel, simple lemma for hypergraph cycle analysis, enabling the proof of a conjecture and providing new bounds on cycle lengths and Turán numbers.

Findings

Proved that hypergraphs with average degree 5r(k+1) contain Berge cycles of k consecutive lengths.

Established that hypergraphs with average degree 5 \, ext{Omega}(k^{r-1}) contain Berge cycles of k consecutive lengths.

Improved bounds on Turán numbers and Zarankiewicz numbers for Berge and even cycles.

Abstract

In this paper, we develop a method for studying cycle lengths in hypergraphs. Our method is built on earlier ones used in [21,22,18]. However, instead of utilizing the well-known lemma of Bondy and Simonovits [4] that most existing methods do, we develop a new and very simple lemma in its place. One useful feature of the new lemma is its adaptiveness for the hypergraph setting. Using this new method, we prove a conjecture of Verstra\"ete [37] that for $r\ge 3$, every $r$-uniform hypergraph with average degree $\Omega(k^{r-1})$ contains Berge cycles of $k$ consecutive lengths. This is sharp up to the constant factor. As a key step and a result of independent interest, we prove that every $r$-uniform linear hypergraph with average degree at least $7r(k+1)$ contains Berge cycles of $k$ consecutive lengths. In both of these results, we have additional control on the lengths of the cycles, which therefore also gives us bounds on the Tur\'an numbers of Berge cycles (for even and odd cycles simultaneously). In relation to our main results, we obtain further improvements on the Tur\'an numbers of Berge cycles and the Zarankiewicz numbers of even cycles. We will also discuss some potential further applications of our method.