Vanishing orders, suspensions and zero degree Tur\'an densities

Abstract

For integers $1\le \ell<k$, the $\ell$-degree Tur\'an density $\pi_\ell(F)$ measures the minimum $\ell$-degree threshold that forces a copy of a fixed $k$-uniform hypergraph $F$, generalizing both the classical Tur\'an density $\pi_1$ and the codegree Tur\'an density $\pi_{k-1}$. Motivated by Erd\H{o}s' characterization of $k$-graphs with zero Tur\'an density, we study the structural implications of vanishing $\ell$-degree Tur\'an density. Our main result concerns the case $\ell=2$. We prove that, for every $k\ge3$, if a $k$-graph $F$ satisfies $\pi_2(F)=0$, then $F$ admits a $2$-vanishing order, that is, a global vertex ordering under which all edges align canonically with respect to their pairs. This extends to all uniformities a structural phenomenon previously known for $3$-graphs, and gives a higher-degree analogue of the classical fact that $\pi_1(F)=0$ forces $F$ to be $k$-partite. In particular, the absence of a $2$-vanishing order is a structural obstruction to vanishing $2$-degree Tur\'an density. We also establish a suspension principle connecting consecutive degree parameters. Given a $(k-1)$-graph $F$, let $\mathcal{S}_F$ be the $k$-graph obtained by adding an apex vertex $v$ and replacing each edge $e\in E(F)$ with $v\cup e$. We show that, for $2\le \ell<k$, $\pi_{\ell}(\mathcal{S}_F)=0$ if and only if $\pi_{\ell-1}(F)=0$. This provides a bridge between different degree Tur\'an densities and allows vanishing results to be lifted across uniformities and degree parameters. As an application, we prove that except the classical Tur\'an density, all other degree Tur\'an densities accumulate at zero. The proof of our main result combines random geometric building blocks, a design-theoretic gluing scheme, and random sparsification to reconcile positive $2$-degree with local vanishing structure.