@article{22184,
  abstract     = {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 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
 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.},
  author       = {Wigderson, Yuval},
  issn         = {0303-1179},
  journal      = {Astérisque},
  pages        = {85--138},
  publisher    = {Societe Mathematique de France},
  title        = {{Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos, Griffiths, Morris, and Sahasrabudhe)}},
  doi          = {10.24033/ast.1255},
  year         = {2026},
}