A Smoother Notion of Spread Hypergraphs

TL;DR

This paper introduces a generalized concept of spread hypergraphs that unifies previous notions, enhancing tools for probabilistic combinatorics and threshold analysis in random graphs.

Contribution

It provides a unified framework that generalizes existing spread hypergraph notions, facilitating new approaches in probabilistic combinatorics.

Findings

Unified framework for spread hypergraphs

Applications to thresholds in random graphs

Potential for new combinatorial proofs

Abstract

Alweiss, Lovett, Wu, and Zhang introduced $q$-spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then $q$-spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of $q$-spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a Hamiltonian cycle appears in the random graph $G_{n,p}$. In this paper we give a common generalization of the original notion of $q$-spread hypergraphs and the variant used by Kahn et al.