A note on conjectures of F. Galvin and R. Rado

TL;DR

This paper surveys results related to conjectures by Galvin and Rado on uncountable posets and graphs, discusses their consistency, and extends Rado's conjecture to chordal graphs within a supercompact cardinal framework.

Contribution

It provides a comprehensive survey of the conjectures and proves the relative consistency of extending Rado's conjecture to chordal graphs assuming a supercompact cardinal.

Findings

Rado's conjecture is consistent relative to a supercompact cardinal.

Galvin's conjecture's consistency remains open.

Extension of Rado's conjecture to chordal graphs is relatively consistent.

Abstract

In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H \subseteq G$ with size $\aleph_1$. Todorcevic has shown that Rado's Conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's Conjecture remains open. In this paper, we survey and collect a variety of results related to these two conjectures. We also show that the extension of Rado's conjecture to the class of all chordal graphs is relatively consistent with the existence of a supercompact cardinal.