Edge-statistics beyond $1/e$

TL;DR

This paper explores the maximum proportion of k-vertex subsets inducing exactly  edges in large graphs, showing that for most , the bound can be improved beyond 1/e, especially when  is far from multiples of k.

Contribution

The paper extends the edge-statistics theorem by establishing improved bounds for  values not at the extremal cases, refining previous results and providing near-optimal bounds in certain regimes.

Findings

The edge-statistics bound of 1/e is tight only at specific  values.

For most , the maximum proportion can be strictly less than 1/e.

Strong upper bounds are obtained when  is far from multiples of k.

Abstract

For integers $k$ and $\ell$, let $\operatorname{ind}(k, \ell)$ be the maximum proportion of $k$-vertex subsets of a large graph that induce exactly $\ell$ edges. The edge-statistics theorem (conjectured by Alon-Hefetz-Krivelevich-Tyomkyn, and proved by Kwan-Sudakov-Tran, Fox-Sauermann, and Martinsson-Mousset-Noever-Truji\'c) asserts that, for $k \to \infty$ and $0 < \ell <\binom{k}{2}$, one has $\operatorname{ind}(k, \ell) \le 1/e + o(1)$. We investigate the ''stability'' of this problem: how can one improve this bound under additional assumptions on $\ell$? In particular, the edge-statistics theorem is tight when $\ell\in \{1,k-1,\binom{k}{2}-(k-1),\binom{k}{2}-1\}$; we show that for all other $\ell$, one can replace $1/e$ with a strictly smaller constant. This extends an analogous result of Ueltzen in the setting of graph inducibility. We also obtain a much stronger (and essentially optimal) upper bound on $\operatorname{ind}(k, \ell)$ when $\ell$ is far from a multiple of $k$, refining and extending previous bounds due to Fox and Sauermann.