The codegree Tur\'an density of $3$-uniform tight cycles

TL;DR

This paper proves that large 3-uniform hypergraphs with high minimum pair-degree necessarily contain certain tight cycles, specifically cycles of length 10 and longer, advancing understanding of cycle thresholds in hypergraph theory.

Contribution

It establishes new minimum degree conditions that guarantee the presence of specific tight cycles in large 3-uniform hypergraphs.

Findings

High minimum pair-degree ensures the existence of a 10-cycle.

The same condition guarantees cycles of length at least 19.

Results extend cycle existence thresholds in 3-uniform hypergraphs.

Abstract

Given any $\varepsilon>0$ we prove that every sufficiently large $n$-vertex $3$-graph $H$ where every pair of vertices is contained in at least $(1/3+\varepsilon)n$ edges contains a copy of $C_{10}$, i.e.\ the tight cycle on $10$ vertices. In fact we obtain the same conclusion for every cycle $C_\ell$ with $\ell\geq 19$.