Anticoncentration in Ramsey graphs and a proof of the Erd\H{o}s-McKay conjecture

TL;DR

This paper establishes precise edge distribution control in Ramsey graphs, linking random-like properties with probabilistic analysis, and proves the Erdős-McKay conjecture using advanced mathematical tools.

Contribution

It introduces a novel approach combining additive structure analysis and diverse mathematical tools to study edge statistics in Ramsey graphs, resolving a longstanding conjecture.

Findings

Controlled the distribution of edges in Ramsey graphs.

Resolved the Erdős-McKay conjecture.

Connected properties of Ramsey graphs with probabilistic and Fourier analysis techniques.

Abstract

An $n$-vertex graph is called $C$-Ramsey if it has no clique or independent set of size $C\log_2 n$ (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a $C$-Ramsey graph. This brings together two ongoing lines of research: the study of "random-like" properties of Ramsey graphs and the study of small-ball probabilities for low-degree polynomials of independent random variables. The proof proceeds via an "additive structure" dichotomy on the degree sequence, and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics, and low-rank approximation. One of the consequences of our result is the resolution of an old conjecture of Erd\H{o}s and McKay, for which Erd\H{o}s offered one of his notorious monetary prizes.