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Borel sets of Rado graphs and Ramsey's Theorem

Natasha Dobrinen

Abstract

The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset $\mathcal{X}\subseteq [ω]^ω$, where $[ω]^ω$ is endowed with the metric topology, each infinite subset $X\subseteq ω$ contains an infinite subset $Y\subseteq X$ such that $[Y]^ω$ is either contained in $\mathcal{X}$ or disjoint from $\mathcal{X}$. Kechris, Pestov, and Todorcevic point out in their seminal 2005 paper the dearth of similar results for homogeneous structures. Such results are a necessary step to the larger goal of finding a correspondence between structures with infinite dimensional Ramsey properties and topological dynamics, extending their correspondence between the Ramsey property and extreme amenability. In this article, we prove an analogue of the Galvin-Prikry theorem for the Rado graph. Any such infinite dimensional Ramsey theorem is subject to constraints following from the 2006 work of Laflamme, Sauer, and Vuksanovic. The proof uses techniques developed for the author's work on the Ramsey theory of the Henson graphs as well as some new methods for fusion sequences, used to bypass the lack of a certain amalgamation property enjoyed by the Baire space.

Borel sets of Rado graphs and Ramsey's Theorem

Abstract

The well-known Galvin-Prikry Theorem states that Borel subsets of the Baire space are Ramsey: Given any Borel subset , where is endowed with the metric topology, each infinite subset contains an infinite subset such that is either contained in or disjoint from . Kechris, Pestov, and Todorcevic point out in their seminal 2005 paper the dearth of similar results for homogeneous structures. Such results are a necessary step to the larger goal of finding a correspondence between structures with infinite dimensional Ramsey properties and topological dynamics, extending their correspondence between the Ramsey property and extreme amenability. In this article, we prove an analogue of the Galvin-Prikry theorem for the Rado graph. Any such infinite dimensional Ramsey theorem is subject to constraints following from the 2006 work of Laflamme, Sauer, and Vuksanovic. The proof uses techniques developed for the author's work on the Ramsey theory of the Henson graphs as well as some new methods for fusion sequences, used to bypass the lack of a certain amalgamation property enjoyed by the Baire space.

Paper Structure

This paper contains 9 sections, 22 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Milliken's Theorem and constraints on the infinite dimensional Ramsey theory of Rado graphs
  3. Constraints on infinite dimensional Ramsey theory of Rado graphs
  4. Brief introduction to topological Ramsey spaces
  5. Strong Rado coding trees
  6. A Halpern-Läuchli-style Theorem for strong Rado coding trees
  7. Borel sets of strong Rado coding trees are completely Ramsey
  8. The Main Theorem
  9. Concluding remarks and further directions

Key Result

Theorem 1.1

Given positive integers $m$ and $\ell$, suppose the collection of all $m$-element subsets of $\omega$ is partitioned into $\ell$ pieces. Then there is an infinite subset $N\subseteq\omega$ such that all $m$-element subsets of $N$ are contained in the same piece of the partition.

Theorems & Definitions (61)

  • Theorem 1.1: Infinite Ramsey Theorem, Ramsey30
  • Theorem 1.2: Galvin and Prikry, Galvin/Prikry73
  • Theorem 1.3: Ellentuck, Ellentuck74
  • Remark 1.4
  • Theorem 1.6
  • Definition 2.1
  • Definition 2.2: Strong Subtrees
  • Theorem 2.3: Halpern-Läuchli, Halpern/Lauchli66
  • Definition 2.4: Milliken space
  • Theorem 2.5: Milliken, Milliken81
  • ...and 51 more