{"publication_status":"published","date_published":"2023-08-24T00:00:00Z","abstract":[{"lang":"eng","text":"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.\r\n\r\nThe 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."}],"department":[{"_id":"MaKw"}],"day":"24","file":[{"content_type":"application/pdf","file_name":"2023_ForumMathematics_Kwan.pdf","access_level":"open_access","creator":"dernst","relation":"main_file","checksum":"54b824098d59073cc87a308d458b0a3e","file_id":"14500","success":1,"date_created":"2023-11-07T09:16:23Z","file_size":1218719,"date_updated":"2023-11-07T09:16:23Z"}],"type":"journal_article","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode"},"date_updated":"2023-11-07T09:18:57Z","month":"08","article_processing_charge":"Yes","status":"public","title":"Anticoncentration in Ramsey graphs and a proof of the Erdős–McKay conjecture","quality_controlled":"1","date_created":"2023-11-07T09:02:48Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa":1,"oa_version":"Published Version","scopus_import":"1","publisher":"Cambridge University Press","citation":{"chicago":"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>.","apa":"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>","ista":"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.","ama":"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>","short":"M.A. Kwan, A. Sah, L. Sauermann, M. Sawhney, Forum of Mathematics, Pi 11 (2023).","mla":"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>.","ieee":"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."},"article_type":"original","language":[{"iso":"eng"}],"article_number":"e21","_id":"14499","doi":"10.1017/fmp.2023.17","has_accepted_license":"1","publication":"Forum of Mathematics, Pi","author":[{"id":"5fca0887-a1db-11eb-95d1-ca9d5e0453b3","full_name":"Kwan, Matthew Alan","first_name":"Matthew Alan","last_name":"Kwan","orcid":"0000-0002-4003-7567"},{"last_name":"Sah","first_name":"Ashwin","full_name":"Sah, Ashwin"},{"full_name":"Sauermann, Lisa","last_name":"Sauermann","first_name":"Lisa"},{"full_name":"Sawhney, Mehtaab","first_name":"Mehtaab","last_name":"Sawhney"}],"publication_identifier":{"issn":["2050-5086"]},"external_id":{"arxiv":["2208.02874"]},"file_date_updated":"2023-11-07T09:16:23Z","acknowledgement":"Kwan was supported for part of this work by ERC Starting Grant ‘RANDSTRUCT’ No. 101076777. Sah and Sawhney were supported by NSF Graduate Research Fellowship Program DGE-2141064. Sah was supported by the PD Soros Fellowship. Sauermann was supported by NSF Award DMS-2100157, and for part of this work by a Sloan Research Fellowship.","keyword":["Discrete Mathematics and Combinatorics","Geometry and Topology","Mathematical Physics","Statistics and Probability","Algebra and Number Theory","Analysis"],"ddc":["510"],"project":[{"name":"Randomness and structure in combinatorics","_id":"bd95085b-d553-11ed-ba76-e55d3349be45","grant_number":"101076777"}],"volume":11,"intvolume":"        11","year":"2023"}