Any $k$-graph with zero $\ell$-degree Tur\'an density is layered

Abstract

The codegree Tur\'an density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. Ding, Lamaison, Liu, Wang, and Yang (JLMS, 2025) studied the problem of what 3-graphs $F$ satisfy $\pi_{\mathrm{co}}(F) = 0$. They introduced layered $3$-graphs and conjectured that a $3$-graph has zero codegree Tur\'an density if and only if it is layered and has zero uniform Tur\'an density. For $k\ge 3$, a $k$-graph is called layered if its vertices can be labelled so that every edge has a unique maximum label and two edges with the same maximum label have the same label multiset. In this paper, we show that every non-layered $k$-graph $F$ on $m$ vertices satisfies \[ \pi_{\mathrm{co}}(F)\ge q_{k,m}^{-q_{k,m}}>0, \quad \text{where}\quad q_{k,m}=\frac{(k-1)^{m+1}-1}{k-2}, \] which implies any $k$-graph with zero $\ell$-degree Tur\'an density is layered, and the case $k=3$ confirms the conjecture of Ding, Lamaison, Liu, Wang, and Yang.