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Gaussian random graphs and Ramsey numbers

Zach Hunter, Aleksa Milojević, Benny Sudakov

TL;DR

The paper reframes Ramsey lower bounds through Gaussian random geometric graphs, replacing sphere-based constructions with Gaussian vectors to gain independence and simpler analysis. By bounding red- and blue-clique probabilities and introducing perfect sequences, it achieves exponential lower bounds R(ℓ, Cℓ) ≥ (p_C^{-1/2} + ε)^ℓ and, for large C, sharper constants such as ε ≈ (e^{1/24}-1)p_C^{-1/2}. A coordinate-change via Bartlett decomposition and an inductive framework bound the relevant probabilities, yielding a more transparent route to improved quantitative bounds. This approach enhances the theoretical toolkit for Ramsey-type problems and clarifies how high-dimensional Gaussian structure can tighten combinatorial limits.

Abstract

We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify their analysis significantly. As a consequence of this simpler analysis, we also obtain better quantitative bounds.

Gaussian random graphs and Ramsey numbers

TL;DR

The paper reframes Ramsey lower bounds through Gaussian random geometric graphs, replacing sphere-based constructions with Gaussian vectors to gain independence and simpler analysis. By bounding red- and blue-clique probabilities and introducing perfect sequences, it achieves exponential lower bounds R(ℓ, Cℓ) ≥ (p_C^{-1/2} + ε)^ℓ and, for large C, sharper constants such as ε ≈ (e^{1/24}-1)p_C^{-1/2}. A coordinate-change via Bartlett decomposition and an inductive framework bound the relevant probabilities, yielding a more transparent route to improved quantitative bounds. This approach enhances the theoretical toolkit for Ramsey-type problems and clarifies how high-dimensional Gaussian structure can tighten combinatorial limits.

Abstract

We give a simple proof of the recent remarkable exponential improvement for Ramsey lower bounds, obtained by Ma, Shen and Xie. Our key ingredient is an alternative construction based on Gaussian random graphs, which allows us to simplify their analysis significantly. As a consequence of this simpler analysis, we also obtain better quantitative bounds.

Paper Structure

This paper contains 8 sections, 2 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Gaussian random geometric graph $G(n, d, p)$.
  3. Applying Gaussian random graphs to Ramsey number lower bounds
  4. Organization.
  5. Preliminaries
  6. Perfect sequences
  7. Changing the perspective
  8. Proofs of Preliminaries

Key Result

Theorem 1.1

For any $C>1$, there exists a constant $\varepsilon$ such that, for all sufficiently large $\ell$, we have [0.3] where $p_C\in (0, 1/2)$ is the unique solution to the equation $C=\frac{\log p_C}{\log (1-p_C)}$.

Theorems & Definitions (17)

  • Theorem 1.1
  • proof : Proof of Lemma \ref{['lemma:bookkeeping']} assuming Proposition \ref{['prop:clique_probabilities']}.
  • proof
  • proof : Proof of Theorem \ref{['thm:main']} assuming Proposition \ref{['prop:clique_probabilities']}.
  • Theorem 2.2: Lemma 1 in LM
  • proof : Proof of Proposition \ref{['prop:clique_probabilities']} based on Proposition \ref{['prop:perfect_probabilities']}.
  • proof
  • proof
  • proof
  • proof : Proof of Claim \ref{['claim:auxiliary_1']}.
  • ...and 7 more