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

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