NOT PROVABLE
Open in general, but there exist models of set theory where the result is false.
- $100
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?
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 $\mathfrak{c}=\aleph_2$.
View the LaTeX source
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #474, https://www.erdosproblems.com/474, accessed 2025-11-15
