Vanishing orders and zero degree Tur\'an densities

TL;DR

This paper investigates the structural properties of hypergraphs with zero $ ext{l}$-degree Turán density, revealing a global vertex ordering for $ ext{l}=2$ and showing that $ ext{l}=2$ densities can accumulate at zero.

Contribution

It establishes a structural characterization for hypergraphs with zero 2-degree Turán density, extending classical results and introducing new techniques for analyzing vanishing densities.

Findings

Zero 2-degree Turán density implies a canonical vertex ordering.

$ ext{l}=2$ Turán densities can accumulate at zero, unlike $ ext{l}=1$ densities.

Provides necessary conditions for $ ext{l}$-degree densities to vanish for $3 \,\leq\, \ell \leq k-1$.

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. We prove for every uniformity $k\ge 3$ that if $\pi_2(F)=0$, then $F$ admits a $2$-vanishing order-a global vertex ordering under which all edges align canonically. This provides a higher-degree analogue of the classical fact that $\pi_1(F)=0$ forces $k$-partiteness, and identifies a structural obstruction to vanishing $2$-degree Tur\'an density. As an application, we show that, unlike $\pi_1$, $\pi_2$ accumulates at $0$. For $3\le \ell\le k-1$, we also obtain weaker necessary conditions for $\pi_\ell(F)=0$. The proof combines random geometric building blocks, a design-theoretic gluing scheme, and random sparsification to reconcile positive $2$-degree with local vanishing structure.