PROVED
This has been solved in the affirmative.
For $A\subset \mathbb{R}^2$ we define the upper density as\[\overline{\delta}(A)=\limsup_{R\to \infty}\frac{\lambda(A \cap B_R)}{\lambda(B_R)},\]where $\lambda$ is the Lebesgue measure and $B_R$ is the ball of radius $R$.
Estimate\[m_1=\sup \overline{\delta}(A),\]where $A$ ranges over all measurable subsets of $\mathbb{R}^2$ without two points distance $1$ apart. In particular, is $m_1\leq 1/4$?
A question of Moser
[Mo66]. A lower bound of $m_1\geq \pi/8\sqrt{3}\approx 0.2267$ is given by taking the union of open circular discs of radius $1/2$ at a regular hexagonal lattice suitably spaced apart. Croft
[Cr67] gives a small improvement of $m_1\geq 0.22936$.
The trivial upper bound is $m_1\leq 1/2$, since for any unit vector $u$ the sets $A$ and $A+u$ must be disjoint. Erdős' question was solved by Ambrus, Csiszárik, Matolcsi, Varga, and Zsámboki
[ACMVZ23] who proved that $m_1\leq 0.247$.
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This page was last edited 02 October 2025.
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 #232, https://www.erdosproblems.com/232, accessed 2026-02-13
