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Invited talk

Tuesday, July 20, 14:45 ~ 15:45 UTC-3

Topological Ramsey spaces: a space of partitions

Carlos Di Prisco

Universidad de Los Andes, Colombia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

F. P. Ramsey published in 1930 an article on the decidability of a fragment of first order logic. A central element of the proof is a combinatorial theorem of independent interest that originated a whole branch of combinatorial theory of infinite sets, known today as Ramsey Theory. This theory has been developed substantially during the last fifty years with applications to mathematical analysis and other areas of mathematics.

Carlson and Simpson initiated the study of a dual Ramsey theory centered on partitions of the set of natural numbers, instead of sets of natural numbers. They also proposed a general framework to study similar combinatorial properties shared by a diversity of spaces, collected under the name of Ramsey spaces. The theory of Ramsey spaces was reformulated and expanded by Todorcevic and continued by other researchers.

In this talk we will present several results about the Ramsey space of infinite partitions of the set of natural numbers. In particular, we propose a definition for coideals of this space and present several of their main properties complementing research initiated by Matet and Halbeisen.

Bibliography.

Carlson, T. J. and S. G. Simpson, A dual form of Ramsey's theorem. Adv. in Math. 53 (1984), no. 3, 265–290.

Matet, P., Partitions and filters. Journal of Symbolic Logic 51 (1986) 12-21.

Halbeisen, L., Ramseyan ultrafilters. Fundamenta Mathematicae 169 (2001) 233-248.

Todorcevic, S., Introduction to Ramsey spaces. Princeton University Press, Princeton, New Jersey, 2010.

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