Oscillating subalgebras of the atomless countable Boolean algebra

TL;DR

This work identifies an infinite big Ramsey degree phenomenon for the countable atomless Boolean algebra by focusing on the $3$-atom subalgebras. It provides an explicit interval-algebra representation of the countable atomless Boolean algebra $oldsymbol{B}$ via the rational interval set $X=[0,1)\cap\mathbb{Q}$ and defines a novel oscillation-based coloring using $ ext{int}(a)$ and $ ext{osc}(a_0,a_1)$. A coloring $oldsymbol{ ext{Emb}}(oldsymbol{B}_3,oldsymbol{B}) ooldsymbol{ u}$ is built from the oscillation of atoms, and it is shown that for every countable atomless subalgebra $oldsymbol{C}$ and every $n$, one can realize oscillation $n$ between disjoint elements of $oldsymbol{C}$, ensuring every color occurs within every such $oldsymbol{C}$. Consequently, the big Ramsey degree of $oldsymbol{B}_3$ in $oldsymbol{B}$ is infinite, marking a qualitative divergence from finite-degree cases and guiding future extensions of big Ramsey theory and its dynamical interpretations.

Abstract

We show that the big Ramsey degree of the Boolean algebra with 3 atoms within the countable atomless Boolean algebra is infinite.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.3: Devlin, 1979 devlin1979
• proof
• Definition 3.1
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:main']} using Theorem \ref{['thm:osc']}
• Lemma 3.2
• proof
• Lemma 3.3
• proof