The codegree Turán density of tight cycles
Jie Ma, Mingyuan Rong
TL;DR
This work advances the codegree Turán theory for k-uniform tight cycles by establishing a general 1/3 upper bound for γ(C_ℓ^k) when ℓ is large and k∤ℓ and the smallest prime factor p of k/ gcd(k,ℓ) satisfies p≥3, extending recent 3-uniform results to higher uniformities. The authors introduce the edge-type framework and the (k,ℓ;d)-family construct, enabling new lower bounds γ(C_ℓ^k) ≥ 1/d and, under gcd(k,ℓ)>1, γ(C_ℓ^{k−}) ≥ 1/d; combined with structural upper bounds, this yields exact γ-values for infinitely many pairs (k,ℓ) and, for gcd(k,ℓ)=1, exact values for a natural density φ(k)/k of ℓ. A central structural theorem shows dense k-graphs without a homomorphic C_ℓ^k must admit a global partition with odd edge intersections with a fixed vertex set, which drives the 1/3 bound. The paper also resolves questions about the tightness of earlier lower bounds and extends the results to C_ℓ^{k−}, providing a broad framework with potential applicability to other extremal hypergraph problems. Overall, the work significantly sharpens γ-values for tight cycles across broad arithmetic regimes and clarifies the role of number-theoretic properties in extremal codegree problems.
Abstract
The codegree Turán density $γ(F)$ of a $k$-uniform hypergraph $F$ is the minimum real number $γ\ge 0$ such that every $k$-uniform hypergraph on sufficiently many $n$ vertices, in which every set of $k-1$ vertices is contained in at least $(γ+o(1))n$ edges, contains a copy of $F$. A recent result of Piga, Sanhueza-Matamala, and Schacht determines that $γ(C_{\ell}^3)=\frac13$ 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 Turán density of $k$-uniform tight cycles $C_\ell^k$. We establish improved upper and lower bounds on $γ(C_{\ell}^k)$ for general $\ell$ not divisible by $k$. These results yield the following consequences: 1). For any prime $k \ge 3$, we show that $γ(C_{\ell}^k)=\frac13$ 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 $γ(C_{\ell}^k)$ 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 $γ(C_{\ell}^k)$. 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 $\ell$.