Finiteness classes arising from Ramsey-theoretic statements in set theory without choice

TL;DR

This paper explores the existence and classification of infinite sets that violate Ramsey-theoretic principles in set theory without the Axiom of Choice, providing detailed hierarchy placement.

Contribution

It offers a precise analysis of where Ramsey-theoretic failure sets reside within the hierarchy of Dedekind-finite sets in choice-free set theory.

Findings

Identifies specific classes of Dedekind-finite sets related to Ramsey failures

Provides hierarchy placement for these sets within set theory without choice

Enhances understanding of combinatorial set theory in the absence of the Axiom of Choice

Abstract

We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey's or (appropriately phrased) Hindman's theorem; such sets may exist if one does not assume the Axiom of Choice. We obtain very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets.