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H 1 Borel sets of Rado graphs are Ramsey

Auteurs : Dobrinen, Natasha (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : The Galvin-Prikry theorem states that Borel partitions of the Baire space are Ramsey. Thus, given any Borel subset $\chi$ of the Baire space and an infinite set $N$, there is an infinite subset $M$ of $N$ such that $\left [M \right ]^{\omega }$ is either contained in $\chi$ or disjoint from $\chi$ . In their 2005 paper, Kechris, Pestov and Todorcevic point out the dearth of similar results for homogeneous relational structures. We have attained such a result for Borel colorings of copies of the Rado graph. We build a topological space of copies of the Rado graph, forming a subspace of the Baire space. Using techniques developed for our work on the big Ramsey degrees of the Henson graphs, we prove that Borel partitions of this space of Rado graphs are Ramsey.

    Keywords : Rado graph, Ramsey theory, forcing

    Codes MSC :
    03C15 - Denumerable structures
    03E75 - Applications
    05D10 - Ramsey theory

    Ressources complémentaires :
    https://www.cirm-math.fr/RepOrga/2052/Slides/Dobrinen_Luminy_Sept2019.pdf

      Informations sur la Vidéo

      Langue : Anglais
      Date de publication : 14/10/2019
      Date de captation : 25/09/2019
      Collection : Research talks
      Format : MP4
      Durée : 00:51:19
      Domaine : Logic and Foundations ; Combinatorics
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2019-09-25_Dobrinen.mp4

    Informations sur la rencontre

    Nom de la rencontre : 15th International Luminy Workshop in Set Theory / XVe Atelier international de théorie des ensembles
    Organisateurs de la rencontre : Dzamonja, Mirna ; Velickovic, Boban
    Dates : 23/09/2019 - 27/09/2019
    Année de la rencontre : 2019
    URL Congrès : https://conferences.cirm-math.fr/2052.html

    Citation Data

    DOI : 10.24350/CIRM.V.19563603
    Cite this video as: Dobrinen, Natasha (2019). Borel sets of Rado graphs are Ramsey. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19563603
    URI : http://dx.doi.org/10.24350/CIRM.V.19563603


    Voir aussi

    Bibliographie

    1. DOBRINEN, Natasha. Borel sets of Rado graphs and Ramsey's Theorem. arXiv preprint arXiv:1904.00266, 2019. - https://arxiv.org/abs/1904.00266

    2. DOBRINEN, Natasha. The Ramsey theory of the universal homogeneous triangle-free graph. arXiv preprint arXiv:1704.00220, 2017. - https://arxiv.org/abs/1704.00220

    3. DOBRINEN, Natasha. The Ramsey Theory of Henson graphs. arXiv preprint arXiv:1901.06660, 2019. - https://arxiv.org/abs/1901.06660

    4. ELLENTUCK, Erik. A new proof that analytic sets are Ramsey. The Journal of Symbolic Logic, 1974, vol. 39, no 1, p. 163-165. - https://doi.org/10.2307/2272356

    5. GALVIN, Fred et PRIKRY, Karel. Borel sets and Ramsey's theorem 1. The Journal of Symbolic Logic, 1973, vol. 38, no 2, p. 193-198. - https://doi.org/10.2307/2272055

    6. HALPERN, James D. et LÄUCHLI, Hans. A partition theorem. Transactions of the American Mathematical society, 1966, vol. 124, no 2, p. 360-367. - https://doi.org/10.1090/s0002-9947-1966-0200172-2

    7. LAFLAMME*, Claude, SAUER, Norbert W., et VUKSANOVIC, Vojkan. Canonical partitions of universal structures. Combinatorica, 2006, vol. 26, no 2, p. 183-205. - https://doi.org/10.1007/s00493-006-0013-2

    8. KECHRIS, Alexander S., PESTOV, Vladimir G., et TODORCEVIC, Stevo. Fraïssé limits, Ramsey theory, and topological dynamics of automorphism groups. Geometric and Functional Analysis, 2005, vol. 15, no 1, p. 106-189. - https://doi.org/10.1007/s00039-005-0503-1

    9. MILLIKEN, Keith R. A Ramsey theorem for trees. Journal of Combinatorial Theory, Series A, 1979, vol. 26, no 3, p. 215-237. - https://doi.org/10.1016/0097-3165(79)90101-8

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