The hypergraph removal process

TL;DR

This paper analyzes a hypergraph removal process where edges forming a fixed subhypergraph are randomly deleted until no such subhypergraph remains, establishing the asymptotic number of remaining edges under certain conditions.

Contribution

It proves a precise asymptotic for the leftover edges in the hypergraph removal process, confirming a major folklore conjecture for complete hypergraphs.

Findings

Remaining edges scale as n^{k-1/ρ} with high probability

Results hold for strictly k-balanced hypergraphs and pseudorandom dense hypergraphs

Confirms a longstanding conjecture in hypergraph theory

Abstract

Let $k\geq 2$ and fix a $k$-uniform hypergraph $\mathcal{F}$. Consider the random process that, starting from a $k$-uniform hypergraph $\mathcal{H}$ on $n$ vertices, repeatedly deletes the edges of a copy of $\mathcal{F}$ chosen uniformly at random and terminates when no copies of $\mathcal{F}$ remain. Let $R(\mathcal{H},\mathcal{F})$ denote the number of edges that are left after termination. We show that $R(\mathcal{H},\mathcal{F})=n^{k-1/\rho\pm o(1)}$, where $\rho:=(\lvert E(\mathcal{F})\rvert-1)/(\lvert V(\mathcal{F})\rvert -k)$, holds with high probability provided that $\mathcal{F}$ is strictly $k$-balanced and $\mathcal{H}$ is sufficiently dense with pseudorandom properties. Since we may in particular choose $\mathcal{F}$ and $\mathcal{H}$ to be complete graphs, this confirms the major folklore conjecture in the area in a very strong form.