Problem 474

Under what set theoretic assumptions is it true that $\mathbb{R}^2$ can be $3$ -coloured such that, for every uncountable $A\subseteq \mathbb{R}^2$ , $A^2$ contains a pair of each colour?

Source: erdosproblems.com/474

Set theory
Ramsey theory

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SolvedIndependent

An accepted solution exists. The statement can be neither proved nor disproved from the standard axioms of set theory.

Independent of ZFC, relative to the consistency of ZFC with an Erdős or measurable cardinal: Erdős, Hajnal and Rado 1965 settles the not-disprovable side, since their coloring exists under the continuum hypothesis, which Gödel showed consistent with ZFC, and Shelah 1988 settles the not-provable side. The site labels the problem NOT PROVABLE, on Shelah's result, and records as open whether the coloring can fail when $2^{\aleph_0}=\aleph_2$ [Va99], a narrower question than the Statement. It credits the coloring under the continuum hypothesis to Erdős but does not record that this classical result settles the other side, so this page departs from the site's label. The pending claim Shelah 2026 reads a 2026 preprint as removing the large cardinal.

Er95d: Erdős, Paul, On some problems in combinatorial set theory. Publ. Inst. Math. (Beograd) (N.S.) 57(71) (1995), 61-65.
Sh88: Shelah, Saharon, Was Sierpiński right? I. Israel J. Math. (1988), 355-380.
Sh26: Shelah, Saharon, Consistency of square bracket partition relation. arXiv:2601.02923 (2026), 14 pages; not on the site's list; see Shelah 2026.
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).

Problem 474

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