Hypergraphs of arbitrary uniformity with vanishing codegree Turán density

The codegree Turán density ${\pi }_{\text{co}}(F)$ of a $k$-uniform hypergraph (or $k$-graph) $F$ is the infimum over all $d$ such that a copy of $F$ is contained in any sufficiently large $n$-vertex $k$-graph $G$ with the property that any $(k-1)$-subset of $V(G)$ is contained in at least $dn$ edges. The problem of determining ${\pi }_{\text{co}}(F)$ for a $k$-graph $F$ is in general very difficult when $k\ge 3$, and there were previously very few nontrivial examples of $k$-graphs $F$ for which ${\pi }_{\text{co}}(F)$ was known when $k\ge 4$. In this paper, we prove that $C_{\mathrm{\ell }}^{(k)-}$, the $k$-uniform tight cycle of length $\mathrm{\ell }$ minus an edge, has vanishing codegree Turán density if and only if $\mathrm{\ell }\equiv 0,\pm 1(\mathrm{m}\mathrm{o}\mathrm{d}k)$ when $\mathrm{\ell }\ge k+2$. This generalises a result of Piga, Sales and Schülke, who proved that ${\pi }_{\text{co}}(C_{\mathrm{\ell }}^{(3)-})=0$ when $\mathrm{\ell }\ge 5$. The method used to prove that ${\pi }_{\text{co}}(C_{\mathrm{\ell }}^{(k)-})=0$ when $\mathrm{\ell }\equiv \pm 1(\mathrm{m}\mathrm{o}\mathrm{d}k)$ and $\mathrm{\ell }\ge 2k-1$ in fact gives a rather larger class of $k$-graphs with vanishing codegree Turán density. We also answer a question of Piga and Schülke by proving that another family of $k$-graphs, studied by them, has vanishing codegree Turán density.