The codegree Tur\'an density of tight cycles

TL;DR

This paper investigates the codegree Turán density of k-uniform tight cycles, providing bounds, exact values for certain cases, and addressing open questions, thereby advancing understanding of hypergraph extremal problems.

Contribution

It establishes improved bounds, exact densities for specific cases, and resolves open questions regarding the codegree Turán density of k-uniform tight cycles.

Findings

For prime k ≥ 3, γ(C_ℓ^k)=1/3 for large ℓ not divisible by k.

Exact γ(C_ℓ^k) determined for a set of ℓ with positive density.

Resolved a question on the tightness of previous constructions.

Abstract

The codegree Tur\'an density $\gamma(F)$ of a $k$-uniform hypergraph $F$ is the minimum real number $\gamma \ge 0$ such that every $k$-uniform hypergraph on sufficiently many $n$ vertices, in which every set of $k-1$ vertices is contained in at least $(\gamma+o(1))n$ edges, contains a copy of $F$. A recent result of Piga, Sanhueza-Matamala, and Schacht determines that $\gamma(C_{\ell}^3)=\frac13$ for every $3$-uniform tight cycle $C_\ell^3$ of length $\ell$, where $\ell \ge \ell_0$ and $\ell$ is not divisible by $3$. In this paper, we investigate the codegree Tur\'an density of $k$-uniform tight cycles $C_\ell^k$. We establish improved upper and lower bounds on $\gamma(C_{\ell}^k)$ for general $\ell$ not divisible by $k$. These results yield the following consequences: 1). For any prime $k \ge 3$, we show that $\gamma(C_{\ell}^k)=\frac13$ for all sufficiently large $\ell$ not divisible by $k$, generalizing the above theorem of Piga et al. 2). For all $k \ge 3$, we determine the exact value of $\gamma(C_{\ell}^k)$ for integers $\ell$ not divisible by $k$ in a set of (natural) density at least $\frac{\varphi(k)}{k}$, where $\varphi(\cdot)$ denotes Euler's totient function. 3). We give a complete answer to a question of Han, Lo, and Sanhueza-Matamala concerning the tightness of their construction for $\gamma(C_{\ell}^k)$. Moreover, our results also determine the codegree Tur\'an density of $C_\ell^{k-}$, that is, the $k$-uniform tight cycle of length $\ell$ with one edge removed, for a new set of integers $\ell$ of positive density for every $k \ge 3$. Our upper bound result is based on a structural characterization of $C_{\ell}^k$-free $k$-uniform hypergraphs with high minimum codegree, while the lower bounds are derived from a novel construction model, coupled with the arithmetic properties of the integers $k$ and $\ell$.