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Claim. Problem 474 asks under which set-theoretic assumptions $\mathbb{R}^2$ has a $3$-coloring of its pairs such that every uncountable $A\subseteq\mathbb{R}^2$ contains a pair of each color; in square-bracket notation, when $2^{\aleph_0}\not\to[\aleph_1]^2_3$ holds. Shelah's Theorem 2.1 (§2) proves: if $\mu=\mu^{<\mu}<\lambda$ and $\lambda$ is a strongly inaccessible measurable cardinal above $\mu$, or, for $\mu=\aleph_0$, the least $\lambda$ with $\lambda\to(\omega_1)^{<\omega}_2$ (an Erdős cardinal), then a $\mu$-complete forcing that collapses no cardinal makes $2^\mu=\lambda$ and $\lambda\to[\mu^+]^2_3$ (Claim 2.6: $\lambda\to[\aleph_1]^2_3$ when $\mu=\aleph_0$); indeed, for any $\sigma<\mu$ colors, some set of size $\mu^+$ realizes at most two of them. So, relative to the consistency of ZFC with such a cardinal, it is consistent with ZFC that $2^{\aleph_0}\to[\aleph_1]^2_3$: every $3$-coloring of the pairs of a set of size continuum has an uncountable subset omitting a color, and the coloring Erdős asked for does not exist. ZFC therefore does not prove that the coloring exists, which is the site's label, not provable, and the introduction (p. 356) says that this settles the Erdős--Hajnal problem.

Hypothesis. The unprovability is relative to a large cardinal: the model is built from a strongly inaccessible Erdős cardinal (for $\mu=\aleph_0$), or a measurable one, and the continuum in it is that former large cardinal, so very large. The paper lists the minimal cases it leaves open (p. 356), among them $\aleph_2\to[\aleph_1]^2_3$ and $2^{\aleph_0}\to[\aleph_2]^2_3$; the question the problem page records from [Va99] as open, whether $2^{\aleph_0}\to[\aleph_1]^2_3$ is consistent with $2^{\aleph_0}=\aleph_2$, asks for the first of them in a model where the continuum is $\aleph_2$. A later preprint claiming the same consistency with no large cardinal has its own claim page, Shelah 2026.

The other direction. Under the continuum hypothesis the coloring exists, by Erdős, after Sierpiński and Kurepa for two colors; a published proof is Theorem 17 of Erdős, Hajnal and Rado (1965), the accepted partial claim page Erdős, Hajnal and Rado 1965, and the problem page's Progress paragraph records a transport of Todorcevic's theorem $\aleph_1\not\to[\aleph_1]^2_3$ along a bijection of $\mathbb{R}^2$ with $\omega_1$ as another route. So the existence of the coloring is not refutable in ZFC either: CH suffices for the coloring, while ZFC alone, granted the large cardinal, does not.

Covers. The not-provable side of the problem: relative to the consistency of ZFC with an Erdős or measurable cardinal, ZFC does not prove that the coloring exists. It does not show that ZFC fails to refute the coloring, and one side alone leaves the question open, so on its own this result leaves Problem 474 open. The CH direction above is an accepted partial claim that settles the other side, the not-disprovable one, so the two together make the existence of the coloring independent of ZFC if ZFC with such a cardinal is consistent; the pending claim Shelah 2026 reads a 2026 preprint as removing the large cardinal, which would give the independence with no large cardinal.

Source. Saharon Shelah, Was Sierpiński right? I, Israel J. Math. 62 (1988), no. 3, 355--380, doi:10.1007/BF02783304; received 18 March 1987, in revised form 12 January 1988; the paper is Sh:276 in the author's archive, described on the source card. The issue is dated October 1988 and carries no day, so this page's date is the first of that month. This page states Theorem 2.1, Claim 2.6 and the introduction's remarks; nothing on this page is independently reviewed by this project.

Acceptance. Refereed: the result is a journal paper in the Israel Journal of Mathematics. Reviewed: the curator of erdosproblems.com, T. F. Bloom, labels Problem 474 not provable and credits Shelah's paper with the consistency of $2^{\aleph_0}\to[\aleph_1]^2_3$ without the continuum hypothesis (problem page last edited 1 February 2026).