On the limit of the positive $\ell$-degree Tur\'an problem

TL;DR

This paper investigates the asymptotic behavior of the minimum positive ll-degree in hypergraphs avoiding certain subgraphs, establishing the existence of a limit and linking it to an optimization problem on hypergraphons.

Contribution

It proves the existence of the limit of the normalized minimum positive ll-degree in ll-free hypergraphs and characterizes it via a hypergraphon optimization framework.

Findings

The ratio of the minimum positive ll-degree to inom{n-ll}{k-ll} converges as n approaches infinity.

The limit can be characterized by a natural optimization problem on hypergraphons.

Provides an alternative description of the set of accumulation points of almost extremal hypergraphs.

Abstract

The minimum positive $\ell$-degree $\delta^+_{\ell}(G)$ of a non-empty $k$-graph $G$ is the maximum $m$ such that every $\ell$-subset of $V(G)$ is contained in either none or at least $m$ edges of $G$; let $\delta^+_{\ell}(G):=0$ if $G$ has no edges. For a family $\mathcal F$ of $k$-graphs, let $co^+ex_\ell(n,\mathcal F)$ be the maximum of $\delta^+_{\ell}(G)$ over all $\mathcal F$-free $k$-graphs $G$ on $n$ vertices. We prove that the ratio $co^+ex_\ell(n,\mathcal F)/{n-\ell\choose k-\ell}$ tends to limit as $n\to\infty$, answering a question of Halfpap, Lemons and Palmer. Also, we show that the limit can be obtained as the value of a natural optimisation problem for $k$-hypergraphons; in fact, we give an alternative description of the set of possible accumulation points of almost extremal $k$-graphs.