Claims

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1965_03_01_erdos_hajnal_rado: Theorem 17 of Erdős, Hajnal and Rado (Acta Math. Acad. Sci. Hungar., 1965) gives Problem 474's three-coloring of pairs of reals under CH, which Gödel showed consistent with ZFC, so ZFC does not refute that the coloring exists.

1988_10_01_shelah: Shelah (Israel J. Math., 1988) proved, from a large cardinal, that it is consistent that every three-coloring of pairs of reals has an uncountable set missing a color, so the coloring Erdős asked for is not provable in ZFC.

2026_01_06_shelah: Shelah (arXiv, 2026) claims consistency of positive square-bracket relations below a small continuum with no large cardinal, read in the problem's thread as settling Problem 474 without large cardinal strength.

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