The codegree Turán density of tight cycles

Abstract The codegree Turán density of a ‐uniform hypergraph is the minimum real number such that, for all sufficiently large , every ‐uniform hypergraph on vertices in which every set of vertices is contained in at least edges contains a copy of . A recent result of Piga, Sanhueza‐Matamala, and Schacht determines that for every 3‐uniform tight cycle of length , where and is not divisible by 3. In this paper, we investigate the codegree Turán density of ‐uniform tight cycles . We establish improved upper and lower bounds on for general not divisible by . These results yield the following consequences: For any prime , we show that for all sufficiently large not divisible by , generalizing the above theorem of Piga et al. For all , we determine the exact value of for integers not divisible by in a set of (natural) density at least , where denotes Euler's totient function. We provide a negative answer to a question of Han, Lo, and Sanhueza‐Matamala regarding the codegree covering threshold for tight cycles. Moreover, our results also determine the codegree Turán density of , that is, the ‐uniform tight cycle of length with one edge removed, for a new set of integers of positive density for every . Our upper‐bound result is based on a structural characterization of ‐free ‐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 and .