CIRM - Videos & books Library - Borel sets of Rado graphs are Ramsey

Multi angle

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

00:00

Résumé : The Galvin-Prikry theorem states that Borel partitions of the Baire space are Ramsey. Thus, given any Borel subset

χ

of the Baire space and an infinite set

N

, there is an infinite subset

M

N

such that

{[M]}^{ω}

is either contained in

χ

or disjoint from

χ

. 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

Réalisateur :

Hennenfent, Guillaume

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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

Données de citation

DOI : 10.24350/CIRM.V.19563603
Citer cette vidéo: 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

[ Post-edited] Chain conditions, unbounded colorings and the $C$ -sequence spectrum / Auteur de la Conférence Rinot, Assaf.
[Multi angle] Towards the right generalization of descriptive set theory to uncountable cardinals / Auteur de la Conférence Motto Ros, Luca.
[Multi angle] Lower bounds for the perfect set property at weakly compact cardinals / Auteur de la Conférence Müller, Sandra.
[Multi angle] Some results on set mappings / Auteur de la Conférence Komjáth, Péter.

Bibliographie

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

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

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

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

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

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

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

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

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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