  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.
