---
OA_place: repository
OA_type: green
_id: '22184'
abstract:
- lang: eng
  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: fre
  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."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Yuval
  full_name: Wigderson, Yuval
  id: 2d0023a0-1567-11f0-833d-d5c1e476d4b5
  last_name: Wigderson
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>'
  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>.'
  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.'
  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>.'
  short: Y. Wigderson, Astérisque (2026) 85–138.
date_created: 2026-06-29T12:02:59Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-07-14T09:53:07Z
day: '01'
doi: 10.24033/ast.1255
extern: '1'
external_id:
  arxiv:
  - '2411.09321'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: 'https://doi.org/10.48550/arXiv.2411.09321 '
mathsc:
- 05D10
- 05C55
month: '01'
oa: 1
oa_version: Preprint
page: 85-138
publication: Astérisque
publication_identifier:
  issn:
  - 0303-1179
  - 2492-5926
publication_status: published
publisher: Societe Mathematique de France
quality_controlled: '1'
scopus_import: '1'
status: public
title: 'Exposé Bourbaki 1230 : Upper bounds on diagonal Ramsey numbers (after Campos,
  Griffiths, Morris, and Sahasrabudhe)'
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...