Ramsey’s theorem states that if $N$ is sufficiently large, then no matter how one colors the edges among $N$ vertices with two colors, there are always k 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 $N=k!$ is sufficient, and five years later Erdős and Szekeres improved this bound to $N=4^k$. And then progress stalled for almost 90 years.
In this talk, I will 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 $N=3.993^k$ is sufficient. In particular, I will try to highlight the central role of book graphs, which play an important role in all known approaches to this problem.
[After Campos, Griffiths, Morris, and Sahasrabudhe]
De Olivier Benoist
De Michael Harris
De Alain Connes
De Teresa Krick
De Teresa Krick
![[1227] Le paysage dirigé](/media/cache/video_light/uploads/video/Capture%20d%E2%80%99e%CC%81cran%202025-01-15%20a%CC%80%2012.05.37.png)
![[1228] Lissification de cycles algébriques de petite dimension](/media/cache/video_light/uploads/video/Capture%20d%E2%80%99e%CC%81cran%202025-01-15%20a%CC%80%2011.53.33.png)
![[1229] Valeurs critiques des fonctions $L$ de puissances symétriques de formes modulaire](/media/cache/video_light/uploads/video/1.png)
![[1230] Upper bounds on diagonal Ramsey numbers](/media/cache/video_light/uploads/video/Capture%20d%E2%80%99e%CC%81cran%202025-01-15%20a%CC%80%2012.29.10.png)
![[1237] Moments de fonctions et $L$ stabilité homologique](/media/cache/video_light/uploads/video/Bourbaki.png)