# Dimitrios Vlitas: An inﬁnite self-dual Ramsey theorem

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 .