Problem Statement
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?
Categories:
Set Theory Ramsey Theory
Progress
A problem of Erdős from 1954. Sierpinski and Kurepa independently proved that this is true for $2$-colours. Erdős proved that this is true under the continuum hypothesis that $\mathfrak{c}=\aleph_1$, and offered \$100 for deciding what happens if the continuum hypothesis is not assumed.Shelah proved that it is consistent that the answer is negative, although with a very large value of $\mathfrak{c}$. It remains open whether it is consistent to have a negative answer assuming $\mathfrak{c}=\aleph_2$.
Source: erdosproblems.com/474 | Last verified: January 15, 2026