Hypergraphs with arbitrarily small codegree Tur\'an density

TL;DR

This paper demonstrates that for any small positive number, there exists a hypergraph with arbitrarily small positive codegree Turán density, contrasting with classical Turán density limitations.

Contribution

It constructs hypergraphs with arbitrarily small positive codegree Turán density, showing the density can be made arbitrarily close to zero.

Findings

Existence of hypergraphs with arbitrarily small positive codegree Turán density.

Contrast with classical Turán density limitations.

Provides new insights into hypergraph extremal problems.

Abstract

Let $k\geq 3$. Given a $k$-uniform hypergraph $H$, the minimum codegree $\delta(H)$ is the largest $d\in\mathbb{N}$ such that every $(k-1)$-set of $V(H)$ is contained in at least $d$ edges. Given a $k$-uniform hypergraph $F$, the codegree Tur\'an density $\gamma(F)$ of $F$ is the smallest $\gamma \in [0,1]$ such that every $k$-uniform hypergraph on $n$ vertices with $\delta(H)\geq (\gamma + o(1))n$ contains a copy of $F$. Similarly as other variants of the hypergraph Tur\'an problem, determining the codegree Tur\'an density of a hypergraph is in general notoriously difficult and only few results are known. In this work, we show that for every $\varepsilon>0$, there is a $k$-uniform hypergraph $F$ with $0<\gamma(F)<\varepsilon$. This is in contrast to the classical Tur\'an density, which cannot take any value in the interval $(0,k!/k^k)$ due to a fundamental result by Erd\H{o}s.