A Jump in the Codegree Tur\'an Densities of Long Tight Cycles

TL;DR

This paper investigates the codegree Turán density of long tight cycles in hypergraphs, revealing a significant drop from 1/2 to 1/3 when certain divisibility conditions are not met, with tight bounds established.

Contribution

It proves a new upper bound of 1/3 for the codegree Turán density in cases where the cycle length and uniformity do not satisfy previous divisibility conditions, extending understanding of hypergraph cycle densities.

Findings

When r/gcd(r,ℓ) is odd, the density is at most 1/3.

The 1/3 bound is tight for infinitely many r and large ℓ.

A group-theoretic approach links Turán theorems and oriented colorings.

Abstract

We study the codegree Tur\'an density of $\mathcal{C}_\ell^r$, the $r$-uniform hypergraph tight cycle of length $\ell$. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is even, then the codegree Tur\'an density of $\mathcal{C}_\ell^r$ is $1/2$. We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Tur\'an density. That is, if $\ell$ is sufficiently large and $r/\gcd(r,\ell)$ is odd, then the codegree Tur\'an density of $\mathcal{C}_\ell^r$ can be at most $1/3$. Moreover, this bound is tight for infinitely many uniformities $r$ and all sufficiently large $\ell$ in the corresponding residue classes modulo $r$. Our proof makes use of a group-theoretic connection between Tur\'an-type theorems for tight cycles and ``oriented colorings'' of the edge set of a hypergraph.