article published yes Matthew Alan Kwan author 5fca0887-a1db-11eb-95d1-ca9d5e0453b30000-0002-4003-7567 Ashwin Sah author Lisa Sauermann author Mehtaab Sawhney author MaKw department Randomness and structure in combinatorics project An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog2n (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 probability 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. In particular, a key ingredient is a new sharpened version of the quadratic Carbery–Wright theorem on small-ball probability for polynomials of Gaussians, which we believe is of independent interest. One of the consequences of our result is the resolution of an old conjecture of Erdős and McKay, for which Erdős reiterated in several of his open problem collections and for which he offered one of his notorious monetary prizes. https://research-explorer.ista.ac.at/download/14499/14500/2023_ForumMathematics_Kwan.pdf application/pdfno https://creativecommons.org/licenses/by/4.0/ Cambridge University Press2023 eng Discrete Mathematics and CombinatoricsGeometry and TopologyMathematical PhysicsStatistics and ProbabilityAlgebra and Number TheoryAnalysis 2050-5086 2208.02874 00112386620000110.1017/fmp.2023.17 11 Kwan MA, Sah A, Sauermann L, Sawhney M. Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture. <i>Forum of Mathematics, Pi</i>. 2023;11. doi:<a href="https://doi.org/10.1017/fmp.2023.17">10.1017/fmp.2023.17</a> M.A. Kwan, A. Sah, L. Sauermann, M. Sawhney, Forum of Mathematics, Pi 11 (2023). M. A. Kwan, A. Sah, L. Sauermann, and M. Sawhney, “Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture,” <i>Forum of Mathematics, Pi</i>, vol. 11. Cambridge University Press, 2023. Kwan, M. A., Sah, A., Sauermann, L., &#38; Sawhney, M. (2023). Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture. <i>Forum of Mathematics, Pi</i>. Cambridge University Press. <a href="https://doi.org/10.1017/fmp.2023.17">https://doi.org/10.1017/fmp.2023.17</a> Kwan MA, Sah A, Sauermann L, Sawhney M. 2023. Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture. Forum of Mathematics, Pi. 11, e21. Kwan, Matthew Alan, et al. “Anticoncentration in Ramsey Graphs and a Proof of the Erdős–McKay Conjecture.” <i>Forum of Mathematics, Pi</i>, vol. 11, e21, Cambridge University Press, 2023, doi:<a href="https://doi.org/10.1017/fmp.2023.17">10.1017/fmp.2023.17</a>. Kwan, Matthew Alan, Ashwin Sah, Lisa Sauermann, and Mehtaab Sawhney. “Anticoncentration in Ramsey Graphs and a Proof of the Erdős–McKay Conjecture.” <i>Forum of Mathematics, Pi</i>. Cambridge University Press, 2023. <a href="https://doi.org/10.1017/fmp.2023.17">https://doi.org/10.1017/fmp.2023.17</a>. 144992023-11-07T09:02:48Z2025-09-09T13:16:15Z