Dimitrios Vlitas: An infinite self-dual Ramsey theorem

*PLEASE NOTE THE SPECIAL TIME: 2PM INSTEAD OF 1:30PM*

Friday, March 16, 2012, from 2pm to 3:30pm
Fields Institute, Room 210

Speaker: Dimitrios Vlitas (Paris)

Title: An infinite self-dual Ramsey theorem

Abstract:

Recall that the classical Ramsey theorem states that given any finite coloring of the set of all K elements subsets of ω there exists of an infinite subset A ⊆ ω where the restriction of the coloring is constant.

The dual form of Ramsey theorem, the Carlson-Simpson Theorem, states that given any finite Borel coloring of the set of all partitions of ω into K many classes, there exists a partition r of ω into ω many classes such that the set of all K partitions of ω resulting by identifying classes of r is monochromatic.

There are also the corresponding finite versions of these results, the finite Ramsey Theorem, and the Graham-Rothschild theorem, respectively. S. Solecki recently proved a self dual theorem that implies simultaneously the finite version of the Ramsey theorem and the Graham-Rothschild theorem. He achieved that by introducing the notion of a connection, which roughly speaking is a labelled partition of L into K many classes, for K and L integers. He then proved that given any positive integers K, L and M there exists N such that for any L coloring of all labelled partitions of N into K many pieces, there exists a labelled partition of M into K pieces, such that the set of all labelled partitions of N into M composed with the particular labelled partition of M into K is monochromatic.

The composition is defined in the most natural way by composing partitions, namely that partition N into M pieces and then M into K pieces, so we finally partition N into K. The composition of the label functions is done in the reverse order.

We extend canonically his notion of connection to labeled partitions of ω, with finite or infinitely many classes and we prove the following:

Theorem. For any finite Borel coloring of al l label led K-partitions of ω there is a fixed label led ω-partition of ω such that the set of al l of its reductions, i.e. label led K-partitions of ω which result from putting pieces of the fixed partition together, is monochromatic.

The proof is done by induction on K and the use of the left variable Hales-Jewett Theorem. In the final section of the paper we extend this result by building the corresponding topological Ramsey space Fω,ω .

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