On the thresholds of degenerate hypergraphs

TL;DR

This paper investigates the emergence thresholds of certain degenerate hypergraphs with bounded degree in random hypergraph models, providing improved bounds on when these structures typically appear.

Contribution

The authors establish that the threshold for the appearance of $(d,eta)$-degenerate hypergraphs with bounded degree is at most $n^{-1/d}$, refining previous bounds.

Findings

Threshold for degenerate hypergraphs is at most $n^{-1/d}$

Improves upon previous bounds by Riordan and Kelly-M"uyesser-Pokrovskiy

Applicable to hypergraphs including planar graphs and powers of cycles

Abstract

An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have $e\left(G[U]\right)\le{d\left(|U|-1\right)-\alpha}$. These hypergraphs include many natural graph classes, such as the degenerate hypergraphs, the planar graphs, and the power of cycles. In this paper, we consider the threshold of the emergence of a $(d,\alpha)$-degenerate hypergraph with bounded maximum degree in the Erd\H{o}s-R\'enyi model. We show that its threshold is at most $n^{-1/d}$, improving previous results of Riordan and Kelly-M\"uyesser-Pokrovskiy.