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]
By Olivier Benoist
By Michael Harris
By Alain Connes
By Teresa Krick
By Teresa Krick
By Javier Fresán
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