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arXiv · 2208.02874

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

Matthew Kwan·Ashwin Sah·Lisa Sauermann·Mehtaab Sawhney
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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.

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Matthew Kwan, Ashwin Sah, Lisa Sauermann, Mehtaab Sawhney. 2022-08-04. Anticoncentration in Ramsey graphs and a proof of the Erd\H{o}s-McKay conjecture. https://doi.org/10.1017/fmp.2023.17

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