Claims

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2013_01_03_komjath: Forces a graph of size and chromatic number aleph one with no uncountable infinitely connected subgraph, so a negative answer is consistent; refereed in 2013 and superseded by Soukup's ZFC counterexample.

2014_09_09_soukup: Builds, with no extra axioms, a graph of chromatic number aleph one in which no uncountable set of vertices is infinitely connected, so the problem's question has a negative answer; refereed in 2015 and credited by the site.

2024_02_08_bowler_pitz: Gives a short construction of a graph of chromatic number aleph one in which every uncountable set of vertices has two vertices joined by only finitely many independent paths; refereed in 2025, formalized by others.

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