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A note on conjectures of F. Galvin and R. Rado. (English) Zbl 1315.03077
Summary: 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 $$\aleph_1$$. Todorčević 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.
##### MSC:
 03E05 Other combinatorial set theory 03E35 Consistency and independence results 03E55 Large cardinals 05C15 Coloring of graphs and hypergraphs 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 06A06 Partial orders, general
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