The codegree Tur\'an density of tight cycles

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 $\ell$.