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Big Ramsey combinatorics of the Cantor set and a simple proof of Blass' perfect set theorem

Dragan Mašulović

Abstract

In this paper we present a simple approach to big Ramsey combinatorics of the Cantor set $2^ω$. Using Infinite Dual Ramsey Theorem of Carlson and Simpson, we show that $2^ω$, viewed as a topological space, has finite big Ramsey degrees. We then examine several natural topological first-order structures arising from the Cantor set and prove that each of them inherits finite big Ramsey degrees. As a consequence, we obtain a simple proof of Blass' perfect set theorem, although our method does not recover the sharp bound $(n-1)!$ for the number of colors. We also show that the complete Boolean algebra on countably many atoms has finite big Ramsey degrees, in contrast with the recent result showing that the countable atomless Boolean algebra does not have big Ramsey degrees.

Big Ramsey combinatorics of the Cantor set and a simple proof of Blass' perfect set theorem

Abstract

In this paper we present a simple approach to big Ramsey combinatorics of the Cantor set . Using Infinite Dual Ramsey Theorem of Carlson and Simpson, we show that , viewed as a topological space, has finite big Ramsey degrees. We then examine several natural topological first-order structures arising from the Cantor set and prove that each of them inherits finite big Ramsey degrees. As a consequence, we obtain a simple proof of Blass' perfect set theorem, although our method does not recover the sharp bound for the number of colors. We also show that the complete Boolean algebra on countably many atoms has finite big Ramsey degrees, in contrast with the recent result showing that the countable atomless Boolean algebra does not have big Ramsey degrees.
Paper Structure (5 sections, 8 theorems, 20 equations)

This paper contains 5 sections, 8 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Cantor set has finite big Ramsey degrees
  4. The Cantor set as a topological first-order structure
  5. Open problem.

Key Result

Theorem 1.1

For every perfect subset $P$ of $\mathbb{R}$ and every finite continuous coloring of the set $[P]^n$ of all $n$-element subsets of $P$, there is a perfect set $Q \subseteq P$ such that $[Q]^n$ has at most $(n - 1)!$ colors.

Theorems & Definitions (14)

  • Theorem 1.1: Blass blass-1981
  • Theorem 2.1: Dual Ramsey Theorem carlson-simpson-1984
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof
  • ...and 4 more