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

TL;DR

This paper introduces a straightforward method to analyze big Ramsey degrees of the Cantor set and related structures, providing new proofs and insights into their combinatorial properties and extending to Boolean algebras.

Contribution

It offers a simple proof of Blass' perfect set theorem using big Ramsey theory and demonstrates that several topological structures derived from the Cantor set have finite big Ramsey degrees.

Findings

The Cantor set has finite big Ramsey degrees.

Several topological structures inherit finite big Ramsey degrees.

The Boolean algebra on countably many atoms has finite big Ramsey degrees.

Abstract

In this paper we present a simple approach to big Ramsey combinatorics of the Cantor set $2^\omega$. Using Infinite Dual Ramsey Theorem of Carlson and Simpson, we show that $2^\omega$, 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.