Homogeneous Dual Ramsey Theorem

• Published in: arXiv.org 2019-07
• Main Author: Mijares, Jose G
• Format: Article
• Language: English
• Subjects: Automorphisms; Coloring; Group dynamics; Integers; Numbers; Partitions; Questions; Theorems
• ISSN: 2331-8422

For positive integers \(k < n\) such that \(k\) divides \(n\), let \((n)^k_{\hom}\) be the set of homogeneous \(k\)-partitions of \(\{1, \dots, n\}\), that is, the set of partitions of \(\{1, \dots, n\}\) into \(k\) classes of the same cardinality. In the article "Ramsey properties of infinite measure algebras and topological dynamics of the group of measure preserving automorphisms: some results and an open problem" by Kechris, Sokic, and Todorcevic, the following question was asked: Is it true that given positive integers \(k < m\) and \(N\) such that \(k\) divides \(m\), there exists a number \(n>m\) such that \(m\) divides \(n\), satisfying that for every coloring \((n)^k_{\hom}=C_1\cup\dots\cup C_N\) we can choose \(u\in (n)^m_{\hom}\) such that \(\{t\in (n)^k_{\hom}: t\mbox{ is coarser than } u\}\subseteq C_i\) for some \(i\)? In this note we give a positive answer to that question. This result turns out to be a homogeneous version of the finite Dual Ramsey Theorem of Graham-Rothschild. As explained by Kechris, Sokic, and Todorcevic in their article, our result also proves that the class \(\mathcal{OMBA}_{\mathbb Q_2}\) of naturally ordered finite measure algebras with measure taking values in the dyadic rationals has the Ramsey property.