Claims

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1977_06_01_rodl: Rödl (Proc. Amer. Math. Soc. 1977) proves that large finite chromatic number forces a complete graph or a triangle-free subgraph of large chromatic number, which gives the case m = aleph_0 with r at most 4.

1988_09_01_komjath_shelah: Komjáth and Shelah (J. Symbolic Logic 1988) prove it consistent with ZFC that an aleph_1-chromatic graph has only countably chromatic triangle-free subgraphs, so the statement for every cardinal is not a theorem of ZFC.

2026_08_16_dottedcalculator: A forum proof claim of August 2026 posted under the username DottedCalculator, with a write-up generated by GPT 5.6 Sol, asserting the affirmative answer for m = aleph_0 and every r; AI-written, unreviewed.

2026_09_06_land: Johan Land's September 2026 manuscript constructs a graph of chromatic number aleph_1 all of whose triangle-free subgraphs are countably colorable, answering the question negatively; AI-assisted, Lean not built.

2026_09_07_steiner: Corollary 1.4 of Steiner's preprint (arXiv, September 2026): every infinite graph of infinite chromatic number has a subgraph of infinite chromatic number and odd girth at least g, for every odd g; the case m = aleph_0.

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