The codegree Turán density of tight cycles

Abstract

The codegree Turán 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 $\gamma (C_{\ell }^3)=\frac{1}{3}$ 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 Turán 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 $\gamma (C_{\ell }^k)=\frac{1}{3}$ 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 $\frac{\phi (k)}{k}$, where $\phi (\cdot )$ 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 Turán 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 $\ell$.

32 pages, 3 figures