The edge-statistics conjecture for hypergraphs

TL;DR

This paper proves that for large hypergraphs, the fraction of k-vertex subsets with a specific number of edges (not 0 or complete) is at most 1/e, confirming a hypergraph version of the edge-statistics conjecture.

Contribution

It establishes an essentially optimal upper bound on the fraction of k-vertex subsets with a given number of edges, generalizing previous limited results and answering a conjecture.

Findings

Fraction is at most 1/e + ε for large hypergraphs.

Result applies to all non-trivial edge counts, not just special cases.

Provides a stronger bound when the edge count is far from 0 or maximum.

Abstract

Let $r,k,\ell$ be integers such that $0\le\ell\le\binom{k}{r}$. Given a large $r$-uniform hypergraph $G$, we consider the fraction of $k$-vertex subsets which span exactly $\ell$ edges. If $\ell$ is 0 or $\binom{k}{r}$, this fraction can be exactly 1 (by taking $G$ to be empty or complete), but for all other values of $\ell$, one might suspect that this fraction is always significantly smaller than 1. In this paper we prove an essentially optimal result along these lines: if $\ell$ is not 0 or $\binom{k}{r}$, then this fraction is at most $(1/e) + \varepsilon$, assuming $k$ is sufficiently large in terms of $r$ and $\varepsilon>0$, and $G$ is sufficiently large in terms of $k$. Previously, this was only known for a very limited range of values of $r,k,\ell$ (due to Kwan-Sudakov-Tran, Fox-Sauermann, and Martinsson-Mousset-Noever-Truji\'{c}). Our result answers a question of Alon-Hefetz-Krivelevich-Tyomkyn, who suggested this as a hypergraph generalisation of their "edge-statistics conjecture". We also prove a much stronger bound when $\ell$ is far from 0 and $\binom{k}{r}$.