[{"arxiv":1,"fulldoi":"https://doi.org/10.24033/ast.1255","language":[{"iso":"eng"}],"status":"public","date_published":"2026-01-01T00:00:00Z","scopus_import":"1","publication_identifier":{"issn":["0303-1179","2492-5926"]},"abstract":[{"text":"Ramsey's theorem states that if N\r\n is sufficiently large, then no matter how one colors the edges among N\r\n vertices with two colors, there are always k\r\n vertices spanning edges in only one color. Given this theorem, it is natural to ask \"how large is sufficiently large?\" Ramsey's original proof showed that N=k!\r\n is sufficient, and five years later Erdős and Szekeres improved this bound to N=4^k\r\n. And then progress stalled for almost 90 years.\r\n\r\nIn this survey, I present the history of the problem, and discuss some of the ideas used in the recent breakthrough of Campos–Griffiths–Morris–Sahasrabudhe, who proved that N=3.993^k\r\n is sufficient. In addition, I discuss the subsequent work of Balister, Bollobás, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, and Tiba, who gave an alternative, and more conceptual, proof.","lang":"eng"},{"text":"Le théorème de Ramsey stipule que si N\r\n est suffisamment grand, alors quelle que soit la manière dont l'on colore les arêtes entre N\r\n sommets avec deux couleurs, il y a toujours k\r\n sommets dont les arêtes ne sont colorées que d'une seule couleur. Compte tenu de ce théorème, il est naturel de se demander \"À quel point N\r\n doit être grand ?\" La preuve originale de Ramsey a montré que N=k!\r\n suffit, et cinq ans plus tard, Erdős et Szekeres ont amélioré cette borne à N=4k\r\n. Puis le progrès s'est arrêté pendant près de 90 ans.\r\n\r\nDans cet exposé, je présente l'histoire du problème et je discute certaines idées utilisées dans la percée récente de Campos--Griffiths-Morris--Sahasrabudhe, qui ont prouvé que N=3,993k\r\n suffit. De plus, je discute le travail suivant de Balister, Bollobás, Campos, Griffiths, Hurley, Morris, Sahasrabudhe, et Tiba, qui ont donné une preuve alternative et plus conceptuelle.","lang":"fre"}],"OA_type":"green","date_created":"2026-06-29T12:02:59Z","year":"2026","publication_status":"published","publisher":"Societe Mathematique de France","external_id":{"arxiv":["2411.09321"]},"month":"01","_id":"22184","doi":"10.24033/ast.1255","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2411.09321 "}],"oa":1,"quality_controlled":"1","day":"01","article_type":"original","extern":"1","page":"85-138","citation":{"ama":"Wigderson Y. Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos, Griffiths, Morris, and Sahasrabudhe). <i>Astérisque</i>. 2026:85-138. doi:<a href=\"https://doi.org/10.24033/ast.1255\">10.24033/ast.1255</a>","apa":"Wigderson, Y. (2026). Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos, Griffiths, Morris, and Sahasrabudhe). <i>Astérisque</i>. Societe Mathematique de France. <a href=\"https://doi.org/10.24033/ast.1255\">https://doi.org/10.24033/ast.1255</a>","ista":"Wigderson Y. 2026. Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos, Griffiths, Morris, and Sahasrabudhe). Astérisque., 85–138.","mla":"Wigderson, Yuval. “Exposé Bourbaki 1230 : Upper Bounds on Diagonal Ramsey Numbers (after Campos, Griffiths, Morris, and Sahasrabudhe).” <i>Astérisque</i>, Societe Mathematique de France, 2026, pp. 85–138, doi:<a href=\"https://doi.org/10.24033/ast.1255\">10.24033/ast.1255</a>.","chicago":"Wigderson, Yuval. “Exposé Bourbaki 1230 : Upper Bounds on Diagonal Ramsey Numbers (after Campos, Griffiths, Morris, and Sahasrabudhe).” <i>Astérisque</i>. Societe Mathematique de France, 2026. <a href=\"https://doi.org/10.24033/ast.1255\">https://doi.org/10.24033/ast.1255</a>.","short":"Y. Wigderson, Astérisque (2026) 85–138.","ieee":"Y. Wigderson, “Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos, Griffiths, Morris, and Sahasrabudhe),” <i>Astérisque</i>. Societe Mathematique de France, pp. 85–138, 2026."},"article_processing_charge":"No","title":"Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos, Griffiths, Morris, and Sahasrabudhe)","author":[{"first_name":"Yuval","last_name":"Wigderson","full_name":"Wigderson, Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","publication":"Astérisque","oa_version":"Preprint","date_updated":"2026-07-14T09:53:07Z","mathsc":["05D10","05C55"],"OA_place":"repository"},{"article_processing_charge":"No","author":[{"first_name":"EDEN","last_name":"KUPERWASSER","full_name":"KUPERWASSER, EDEN"},{"last_name":"SAMOTIJ","first_name":"WOJCIECH","full_name":"SAMOTIJ, WOJCIECH"},{"full_name":"Wigderson, Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","first_name":"Yuval","last_name":"Wigderson"}],"title":"On the Kohayakawa–Kreuter conjecture","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":178,"publication":"Mathematical Proceedings of the Cambridge Philosophical Society","date_updated":"2026-07-14T08:38:08Z","oa_version":"Preprint","mathsc":["05C80","05C55","05D10"],"OA_place":"repository","doi":"10.1017/s0305004125000143","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2307.16611","open_access":"1"}],"intvolume":"       178","oa":1,"quality_controlled":"1","day":"28","extern":"1","article_type":"original","page":"293-320","citation":{"ama":"KUPERWASSER E, SAMOTIJ W, Wigderson Y. On the Kohayakawa–Kreuter conjecture. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. 2025;178(3):293-320. doi:<a href=\"https://doi.org/10.1017/s0305004125000143\">10.1017/s0305004125000143</a>","apa":"KUPERWASSER, E., SAMOTIJ, W., &#38; Wigderson, Y. (2025). On the Kohayakawa–Kreuter conjecture. <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0305004125000143\">https://doi.org/10.1017/s0305004125000143</a>","ista":"KUPERWASSER E, SAMOTIJ W, Wigderson Y. 2025. On the Kohayakawa–Kreuter conjecture. Mathematical Proceedings of the Cambridge Philosophical Society. 178(3), 293–320.","mla":"KUPERWASSER, EDEN, et al. “On the Kohayakawa–Kreuter Conjecture.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 178, no. 3, Cambridge University Press, 2025, pp. 293–320, doi:<a href=\"https://doi.org/10.1017/s0305004125000143\">10.1017/s0305004125000143</a>.","chicago":"KUPERWASSER, EDEN, WOJCIECH SAMOTIJ, and Yuval Wigderson. “On the Kohayakawa–Kreuter Conjecture.” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>. Cambridge University Press, 2025. <a href=\"https://doi.org/10.1017/s0305004125000143\">https://doi.org/10.1017/s0305004125000143</a>.","short":"E. KUPERWASSER, W. SAMOTIJ, Y. Wigderson, Mathematical Proceedings of the Cambridge Philosophical Society 178 (2025) 293–320.","ieee":"E. KUPERWASSER, W. SAMOTIJ, and Y. Wigderson, “On the Kohayakawa–Kreuter conjecture,” <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, vol. 178, no. 3. Cambridge University Press, pp. 293–320, 2025."},"publication_identifier":{"issn":["0305-0041"],"eissn":["1469-8064"]},"abstract":[{"lang":"eng","text":"Let us say that a graph G is Ramsey for a tuple (H1, ... , Hr) of graphs if every r-colouring\r\nof the edges of G contains a monochromatic copy of Hi in colour i, for some i ∈ [[r]].\r\nA famous conjecture of Kohayakawa and Kreuter, extending seminal work of Rödl and\r\nRucinski, predicts the threshold at which the binomial random graph ´ Gn,p becomes Ramsey\r\nfor (H1, ... , Hr) asymptotically almost surely.\r\nIn this paper, we resolve the Kohayakawa–Kreuter conjecture for almost all tuples of\r\ngraphs. Moreover, we reduce its validity to the truth of a certain deterministic statement,\r\nwhich is a clear necessary condition for the conjecture to hold. All of our results actually hold in greater generality, when one replaces the graphs H1, ... , Hr by finite families\r\nH1, ... , Hr. Additionally, we pose a natural (deterministic) graph-partitioning conjecture,\r\nwhich we believe to be of independent interest, and whose resolution would imply the\r\nKohayakawa–Kreuter conjecture."}],"OA_type":"green","date_created":"2026-06-29T10:55:00Z","year":"2025","publication_status":"published","publisher":"Cambridge University Press","external_id":{"arxiv":["2307.16611"]},"month":"04","_id":"22168","arxiv":1,"issue":"3","fulldoi":"https://doi.org/10.1017/s0305004125000143","language":[{"iso":"eng"}],"status":"public","date_published":"2025-04-28T00:00:00Z","scopus_import":"1"},{"publisher":"Cambridge University Press","external_id":{"arxiv":["2110.14483"]},"publication_status":"published","month":"05","_id":"22165","publication_identifier":{"eissn":["1469-2163"],"issn":["0963-5483"]},"year":"2023","abstract":[{"lang":"eng","text":"The book graph 𝐵(𝑘)\r\n𝑛 consists of 𝑛 copies of 𝐾𝑘+1 joined along a common 𝐾𝑘. In the prequel to this paper, we studied the diagonal Ramsey number 𝑟⁡(𝐵(𝑘)\r\n𝑛,𝐵(𝑘)\r\n𝑛). Here we consider the natural off-diagonal variant 𝑟⁡(𝐵(𝑘)\r\n𝑐⁢𝑛,𝐵(𝑘)\r\n𝑛) for fixed 𝑐 ∈(0,1]. In this more general setting, we show that an interesting dichotomy emerges: for very small 𝑐, a simple 𝑘-partite construction dictates the Ramsey function and all nearly-extremal colourings are close to being 𝑘-partite, while, for 𝑐 bounded away from 0, random colourings of an appropriate density are asymptotically optimal and all nearly-extremal colourings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of 𝑐.\r\n\r\n"}],"date_created":"2026-06-29T10:53:47Z","OA_type":"green","status":"public","language":[{"iso":"eng"}],"scopus_import":"1","date_published":"2023-05-01T00:00:00Z","arxiv":1,"fulldoi":"https://doi.org/10.1017/s0963548322000360","issue":"3","date_updated":"2026-07-14T08:27:40Z","oa_version":"Preprint","OA_place":"repository","mathsc":["05C55","05D10"],"author":[{"full_name":"Conlon, David","first_name":"David","last_name":"Conlon"},{"first_name":"Jacob","last_name":"Fox","full_name":"Fox, Jacob"},{"last_name":"Wigderson","first_name":"Yuval","full_name":"Wigderson, Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5"}],"title":"Off-diagonal book Ramsey numbers","type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","volume":32,"publication":"Combinatorics, Probability and Computing","article_type":"original","extern":"1","page":"516-545","day":"01","citation":{"apa":"Conlon, D., Fox, J., &#38; Wigderson, Y. (2023). Off-diagonal book Ramsey numbers. <i>Combinatorics, Probability and Computing</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/s0963548322000360\">https://doi.org/10.1017/s0963548322000360</a>","ama":"Conlon D, Fox J, Wigderson Y. Off-diagonal book Ramsey numbers. <i>Combinatorics, Probability and Computing</i>. 2023;32(3):516-545. doi:<a href=\"https://doi.org/10.1017/s0963548322000360\">10.1017/s0963548322000360</a>","ista":"Conlon D, Fox J, Wigderson Y. 2023. Off-diagonal book Ramsey numbers. Combinatorics, Probability and Computing. 32(3), 516–545.","short":"D. Conlon, J. Fox, Y. Wigderson, Combinatorics, Probability and Computing 32 (2023) 516–545.","chicago":"Conlon, David, Jacob Fox, and Yuval Wigderson. “Off-Diagonal Book Ramsey Numbers.” <i>Combinatorics, Probability and Computing</i>. Cambridge University Press, 2023. <a href=\"https://doi.org/10.1017/s0963548322000360\">https://doi.org/10.1017/s0963548322000360</a>.","mla":"Conlon, David, et al. “Off-Diagonal Book Ramsey Numbers.” <i>Combinatorics, Probability and Computing</i>, vol. 32, no. 3, Cambridge University Press, 2023, pp. 516–45, doi:<a href=\"https://doi.org/10.1017/s0963548322000360\">10.1017/s0963548322000360</a>.","ieee":"D. Conlon, J. Fox, and Y. Wigderson, “Off-diagonal book Ramsey numbers,” <i>Combinatorics, Probability and Computing</i>, vol. 32, no. 3. Cambridge University Press, pp. 516–545, 2023."},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2110.14483"}],"doi":"10.1017/s0963548322000360","keyword":["Ramsey theory","book graphs","Ramsey goodness"],"quality_controlled":"1","intvolume":"        32","oa":1},{"status":"public","language":[{"iso":"eng"}],"scopus_import":"1","date_published":"2021-01-01T00:00:00Z","arxiv":1,"fulldoi":"https://doi.org/10.4310/joc.2021.v12.n1.a1","issue":"1","publisher":"International Press of Boston","external_id":{"arxiv":["1912.08328"]},"publication_status":"published","month":"01","_id":"22161","publication_identifier":{"issn":["2156-3527"],"eissn":["2150-959X"]},"year":"2021","abstract":[{"lang":"eng","text":"Recently, Souza introduced blowup Ramsey numbers as a gener-\r\nalization of bipartite Ramsey numbers. For graphs G and H, say\r\nG r\r\n−→ H if every r-edge-coloring of G contains a monochromatic\r\ncopy of H. Let H[t] denote the t-blowup of H. Then the blowup\r\nRamsey number of G, H, r, and t is defined as the minimum n\r\nsuch that G[n] r\r\n−→ H[t]. Souza proved upper and lower bounds on\r\nn that are exponential in t, and conjectured that the exponential\r\nconstant does not depend on G. We prove that the dependence on\r\nG in the exponential constant is indeed unnecessary, but conjecture\r\nthat some dependence on G is unavoidable.\r\nAn important step in both Souza’s proof and ours is a theorem of\r\nNikiforov, which says that if a graph contains a constant fraction\r\nof the possible copies of H, then it contains a blowup of H of\r\nlogarithmic size. We also provide a new proof of this theorem with\r\na better quantitative dependence."}],"date_created":"2026-06-29T10:52:13Z","OA_type":"green","extern":"1","article_type":"original","page":"1-15","day":"01","citation":{"ista":"Fox J, Luo S, Wigderson Y. 2021. Extremal and Ramsey results on graph blowups. Journal of Combinatorics. 12(1), 1–15.","apa":"Fox, J., Luo, S., &#38; Wigderson, Y. (2021). Extremal and Ramsey results on graph blowups. <i>Journal of Combinatorics</i>. International Press of Boston. <a href=\"https://doi.org/10.4310/joc.2021.v12.n1.a1\">https://doi.org/10.4310/joc.2021.v12.n1.a1</a>","ama":"Fox J, Luo S, Wigderson Y. Extremal and Ramsey results on graph blowups. <i>Journal of Combinatorics</i>. 2021;12(1):1-15. doi:<a href=\"https://doi.org/10.4310/joc.2021.v12.n1.a1\">10.4310/joc.2021.v12.n1.a1</a>","ieee":"J. Fox, S. Luo, and Y. Wigderson, “Extremal and Ramsey results on graph blowups,” <i>Journal of Combinatorics</i>, vol. 12, no. 1. International Press of Boston, pp. 1–15, 2021.","chicago":"Fox, Jacob, Sammy Luo, and Yuval Wigderson. “Extremal and Ramsey Results on Graph Blowups.” <i>Journal of Combinatorics</i>. International Press of Boston, 2021. <a href=\"https://doi.org/10.4310/joc.2021.v12.n1.a1\">https://doi.org/10.4310/joc.2021.v12.n1.a1</a>.","short":"J. Fox, S. Luo, Y. Wigderson, Journal of Combinatorics 12 (2021) 1–15.","mla":"Fox, Jacob, et al. “Extremal and Ramsey Results on Graph Blowups.” <i>Journal of Combinatorics</i>, vol. 12, no. 1, International Press of Boston, 2021, pp. 1–15, doi:<a href=\"https://doi.org/10.4310/joc.2021.v12.n1.a1\">10.4310/joc.2021.v12.n1.a1</a>."},"main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.1912.08328"}],"doi":"10.4310/joc.2021.v12.n1.a1","quality_controlled":"1","intvolume":"        12","oa":1,"date_updated":"2026-07-08T10:41:31Z","oa_version":"Preprint","OA_place":"repository","mathsc":["05C35","05C55"],"title":"Extremal and Ramsey results on graph blowups","author":[{"last_name":"Fox","first_name":"Jacob","full_name":"Fox, Jacob"},{"first_name":"Sammy","last_name":"Luo","full_name":"Luo, Sammy"},{"id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5","full_name":"Wigderson, Yuval","last_name":"Wigderson","first_name":"Yuval"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"journal_article","article_processing_charge":"No","volume":12,"publication":"Journal of Combinatorics"},{"publication":"The Electronic Journal of Combinatorics","volume":27,"type":"journal_article","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","title":"Multicolor Ramsey numbers via pseudorandom graphs","author":[{"full_name":"He, Xiaoyu","first_name":"Xiaoyu","last_name":"He"},{"last_name":"Wigderson","first_name":"Yuval","full_name":"Wigderson, Yuval","id":"2d0023a0-1567-11f0-833d-d5c1e476d4b5"}],"article_processing_charge":"No","OA_place":"repository","mathsc":["05C55","05D10"],"date_updated":"2026-07-14T09:42:53Z","oa_version":"Preprint","quality_controlled":"1","oa":1,"intvolume":"        27","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.1910.06287","open_access":"1"}],"doi":"10.37236/9071","citation":{"ista":"He X, Wigderson Y. 2020. Multicolor Ramsey numbers via pseudorandom graphs. The Electronic Journal of Combinatorics. 27(1), P1.32.","apa":"He, X., &#38; Wigderson, Y. (2020). Multicolor Ramsey numbers via pseudorandom graphs. <i>The Electronic Journal of Combinatorics</i>. The Electronic Journal of Combinatorics. <a href=\"https://doi.org/10.37236/9071\">https://doi.org/10.37236/9071</a>","ama":"He X, Wigderson Y. Multicolor Ramsey numbers via pseudorandom graphs. <i>The Electronic Journal of Combinatorics</i>. 2020;27(1). doi:<a href=\"https://doi.org/10.37236/9071\">10.37236/9071</a>","ieee":"X. He and Y. Wigderson, “Multicolor Ramsey numbers via pseudorandom graphs,” <i>The Electronic Journal of Combinatorics</i>, vol. 27, no. 1. The Electronic Journal of Combinatorics, 2020.","chicago":"He, Xiaoyu, and Yuval Wigderson. “Multicolor Ramsey Numbers via Pseudorandom Graphs.” <i>The Electronic Journal of Combinatorics</i>. The Electronic Journal of Combinatorics, 2020. <a href=\"https://doi.org/10.37236/9071\">https://doi.org/10.37236/9071</a>.","short":"X. He, Y. Wigderson, The Electronic Journal of Combinatorics 27 (2020).","mla":"He, Xiaoyu, and Yuval Wigderson. “Multicolor Ramsey Numbers via Pseudorandom Graphs.” <i>The Electronic Journal of Combinatorics</i>, vol. 27, no. 1, P1.32, The Electronic Journal of Combinatorics, 2020, doi:<a href=\"https://doi.org/10.37236/9071\">10.37236/9071</a>."},"extern":"1","article_type":"original","day":"01","year":"2020","date_created":"2026-06-29T12:06:14Z","OA_type":"green","abstract":[{"lang":"eng","text":"A weakly optimal Ks-free (n,d,λ)-graph is ad-regular Ks-free graph on n vertices with d= Θ(n1−α) and spectral expansion λ= Θ(n1−(s−1)α),  for some fixed α >0.  Such a graph is called optimal if additionally α=12s−3.  We prove that if s1,...,sk>3 are fixed positive integers and weakly optimal Ksi-free pseudorandom graphs exist for each 1<i<k, then the multicolor Ramsey numbers satisfy \r\n\r\nΩ(tS+1log2St)6r(s1,...,sk,t)6O(tS+1logSt),\r\nas t→∞, where S=∑ki=1(si−2). This generalizes previous results of Mubayi andVerstraete, who proved the case k= 1, and Alon and Rodl, who proved the cases1=···=sk= 3.  Both previous results used the existence of optimal rather than weakly optimal Ksi-free graphs"}],"publication_identifier":{"eissn":["1077-8926"]},"month":"02","_id":"22185","external_id":{"arxiv":["1910.06287"]},"publisher":"The Electronic Journal of Combinatorics","publication_status":"published","fulldoi":"https://doi.org/10.37236/9071","issue":"1","arxiv":1,"scopus_import":"1","article_number":"P1.32","date_published":"2020-02-01T00:00:00Z","status":"public","language":[{"iso":"eng"}]}]