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Claim. Theorem 17 (p. 145) of P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, states $\aleph_{\alpha+1}\not\to[\aleph_{\alpha+1}]^2_{\aleph_{\alpha+1}}$ for every ordinal $\alpha$; the paper marks it $(*)$, its sign for results proved under the generalized continuum hypothesis. For $\alpha=0$ the proof (pp. 145--146) uses only $2^{\aleph_0}=\aleph_1$, to enumerate the countable subsets of $\omega_1$ in order type $\omega_1$. So under CH the pairs of a set of size $\aleph_1$, hence of $\mathbb{R}$, have a coloring with $\aleph_1$ colors in which every uncountable subset contains a pair of every color. Merging the colors into three nonempty classes gives a $3$-coloring of the pairs of reals in which every uncountable set contains a pair of each color, the coloring Problem 474 asks for; in the problem's notation, CH implies $2^{\aleph_0}\not\to[\aleph_1]^2_3$.

Hypothesis. The continuum hypothesis. It is not a theorem of ZFC, and the accepted claim page Shelah 1988 shows, relative to a large cardinal, that ZFC alone does not prove that the coloring exists; but the hypothesis is consistent with ZFC, as Gödel showed, so the theorem settles the problem's not-disprovable side. With Shelah's result it shows that CH suffices for the coloring, while ZFC alone, granted the large cardinal, does not.

Covers. The not-disprovable side of Problem 474: the coloring exists in every model of ZFC with the continuum hypothesis, and that hypothesis holds in Gödel's constructible universe, so it is consistent with ZFC if ZFC is consistent; ZFC therefore does not refute the existence of the coloring. That side alone leaves the question open. Together with Shelah 1988, which settles the not-provable side relative to the consistency of ZFC with an Erdős or measurable cardinal, it shows that the existence of the coloring is independent of ZFC if ZFC with such a cardinal is consistent.

Source. P. Erdős, A. Hajnal and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), no. 1--2, 93--196, doi:10.1007/BF01886396; the issue is dated March 1965, and this page's date is the first of that month. Erdős and Hajnal's 1971 problem list, Unsolved problems in set theory (§4.2, p. 25), cites Theorem 17 of this paper for the consequence. The site credits the coloring under CH to Erdős, and in [Er95d] (On some problems in combinatorial set theory, Publ. Inst. Math. (Beograd) (N.S.) 57(71) (1995), p. 64) Erdős writes of having proved the affirmative answer under $\mathfrak{c}=\aleph_1$ after posing the question in 1954. No publication of that proof earlier than this paper is recorded here.

Acceptance. Refereed: Acta Mathematica Academiae Scientiarum Hungaricae, volume 16, issue 1--2. The site labels Problem 474 NOT PROVABLE on Shelah's consistency result, so its remark crediting Erdős with the CH case is commentary, not acceptance of this claim, and no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.