Rado's conjecture and its Baire version

TL;DR

This paper explores the consistency and implications of Rado's Conjecture and its Baire version, demonstrating their compatibility with certain forcing axioms and analyzing their impact on reflection principles and partition relations.

Contribution

It establishes the consistency of the Baire Rado's Conjecture with a fragment of PFA and investigates its consequences and limitations in set theory.

Findings

Baire Rado's Conjecture is compatible with a fragment of PFA.

Baire Rado's Conjecture does not imply Rado's Conjecture.

The influence of Rado's Conjecture on polarized partition relations was analyzed.

Abstract

Rado's Conjecture is a compactness/reflection principle that says any nonspecial tree of height $\omega_1$ has a nonspecial subtree of size $\leq \aleph_1$. Though incompatible with Martin's Axiom, Rado's Conjecture turns out to have many interesting consequences that are consequences of forcing axioms. In this paper, we obtain consistency results concerning Rado's Conjecture and its Baire version. In particular, we show a fragment of PFA, that is the forcing axiom for \emph{Baire Indestructibly proper forcings}, is compatible with the Baire Rado's Conjecture. As a corollary, Baire Rado's Conjecture does not imply Rado's Conjecture. Then we discuss the strength and limitations of the Baire Rado's Conjecture regarding its interaction with simultaneous stationary reflection and some families of weak square principles. Finally we investigate the influence of the Rado's Conjecture on some polarized partition relations.