The codegree Tur\'an density of tight cycles minus one edge

TL;DR

This paper proves that large 3-uniform hypergraphs with high minimum pair degree necessarily contain a nearly complete tight cycle of specified length, improving previous bounds in hypergraph cycle theory.

Contribution

It establishes a new minimum degree condition ensuring the presence of a tight cycle minus one edge in large 3-uniform hypergraphs, advancing understanding of hypergraph cycle thresholds.

Findings

High minimum pair degree guarantees the existence of $C_ ext{ell}^{-}$.

Improves previous bounds on cycle containment in hypergraphs.

Results hold for sufficiently large hypergraphs.

Abstract

Given $\alpha>0$ and an integer $\ell\geq5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_\ell^{-}$, a tight cycle on $\ell$ vertices minus one edge. This improves a previous result by Balogh, Clemen, and Lidick\'y.