The edge-statistics conjecture for $\ell \ll k^{6/5}$

TL;DR

This paper proves a bound on the fraction of k-vertex subsets with exactly  edges in large graphs for   k^{6/5}, confirming a conjecture related to edge distribution in graphs.

Contribution

It establishes the edge-statistics conjecture for   k^{6/5}, resolving a long-standing open problem in graph theory.

Findings

Fraction of k-vertex subsets with exactly  edges is at most 1/e + o_k(1)

Confirms the edge-statistics conjecture for   o_k(k^{6/5})

Complements recent results to settle a conjecture of Alon et al.

Abstract

Let $k$ and $\ell$ be positive integers. We prove that if $1 \leq \ell \leq o_k(k^{6/5})$, then in every large enough graph $G$, the fraction of $k$-vertex subsets that induce exactly $\ell$ edges is at most $1/e + o_k(1)$. Together with a recent result of Kwan, Sudakov, and Tran, this settles a conjecture of Alon, Hefetz, Krivelevich, and Tyomkyn.