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Edge-statistics beyond $1/e$

Alexandr Grebennikov, Matthew Kwan

TL;DR

This work sharpens the edge-inducibility story beyond the classical 1/e barrier by proving stability and sharper bounds for ℓ in relation to k. It develops a robust toolkit built around polynomials on the slice of the Boolean cube, including linear and quadratic Littlewood–Offord anticoncentration, transfer principles between slice and product measures, and antichain arguments, to derive asymptotically tight bounds in regimes where ℓ is far from or close to a small multiple of k. The results extend to uniform hypergraphs in a weaker but structurally precise form and are complemented by a computational approach for the smallest-ℓ corner. The paper also proposes set-system analogues and tightness discussions, outlining conjectures and directions for future work on edge-statistics beyond 1/e.

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-Trujić) 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.

Edge-statistics beyond $1/e$

TL;DR

This work sharpens the edge-inducibility story beyond the classical 1/e barrier by proving stability and sharper bounds for ℓ in relation to k. It develops a robust toolkit built around polynomials on the slice of the Boolean cube, including linear and quadratic Littlewood–Offord anticoncentration, transfer principles between slice and product measures, and antichain arguments, to derive asymptotically tight bounds in regimes where ℓ is far from or close to a small multiple of k. The results extend to uniform hypergraphs in a weaker but structurally precise form and are complemented by a computational approach for the smallest-ℓ corner. The paper also proposes set-system analogues and tightness discussions, outlining conjectures and directions for future work on edge-statistics beyond 1/e.

Abstract

For integers and , let be the maximum proportion of -vertex subsets of a large graph that induce exactly edges. The edge-statistics theorem (conjectured by Alon-Hefetz-Krivelevich-Tyomkyn, and proved by Kwan-Sudakov-Tran, Fox-Sauermann, and Martinsson-Mousset-Noever-Trujić) asserts that, for and , one has . We investigate the ''stability'' of this problem: how can one improve this bound under additional assumptions on ? In particular, the edge-statistics theorem is tight when ; we show that for all other , one can replace 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 when is far from a multiple of , refining and extending previous bounds due to Fox and Sauermann.

Paper Structure

This paper contains 15 sections, 26 theorems, 150 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Strong bounds when is far from a small multiple of
  3. Asymptotic results when is close to a small multiple of
  4. Edge-inducibilities in uniform hypergraphs
  5. Proof ideas
  6. Polynomials on a slice of the Boolean hypercube
  7. $\ell$ is far from a multiple of $k$
  8. $\ell$ is close to a nonzero multiple of $k$
  9. Proof of \ref{['better-than-3/4']}
  10. $\ell$ is close to zero
  11. Hypergraphs
  12. Concluding remarks
  13. Edge-statistics for set systems
  14. Tightness of : constructions and discussion
  15. Proofs of \ref{['sparse-polynomials-general', 'dense-polynomials']}

Key Result

Theorem 1.1

Let $k, \ell$ be positive integers such that $k$ is sufficiently large, and $1 \leqslant \ell \leqslant \frac{1}{2}\binom{k}{2}$, $\ell \notin \{1, k-1\}$. Then,

Figures (2)

  • Figure 1: This cartoon plot summarises the upper bounds on $\mathop{\mathrm{ind}}\nolimits(k, \ell)$ given by \ref{['bulk', 'far-from-multiple']}.
  • Figure 2: This cartoon plot summarises the upper bounds on $\mathop{\mathrm{ind}}\nolimits(k, \ell)$ in the range $\ell = O(k)$ and $k \to \infty$ given by \ref{['better-than-1/e', 'far-from-multiple', 'close-to-multiple', '2/e^2']}.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2: Kwan--Sauermann kwan-sauermann-23
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2: Kwan--Sudakov--Tran kwan-sudakov-tran-19
  • Proposition 2.3
  • proof
  • ...and 52 more