A jump in the codegree Turán densities of long tight cycles

Abstract We study the codegree Turán density of $\mathcal{C}_\ell ^{r}$, the $r$-uniform hypergraph tight cycle of length $\ell $. A result of Han, Lo, and Sanhueza-Matamala states that if $\ell $ is sufficiently large and $r/\gcd (r,\ell )$ is even, then the codegree Turán density of $\mathcal{C}_\ell ^{r}$ is $1/2$. We prove that whenever the latter assumption is not satisfied, there is a significant drop in the codegree Turán density. That is, if $\ell $ is sufficiently large and $r/\gcd (r,\ell )$ is odd, then the codegree Turán density of $\mathcal{C}_\ell ^{r}$ can be at most $1/3$. Moreover, this bound is tight for infinitely many uniformities $r$ and all sufficiently large $\ell $ in the corresponding residue classes modulo $r$. Our proof makes use of a group-theoretic connection between Turán-type theorems for tight cycles and “oriented colorings” of the edge set of a hypergraph.