PROVED
This has been solved in the affirmative.
Let $N(X,\delta)$ denote the maximum number of points $P_1,\ldots,P_n$ which can be chosen in a circle of radius $X$ such that\[\| \lvert P_i-P_j\rvert \| \geq \delta\]for all $1\leq i<j\leq n$. (Here $\|x\|$ is the distance from $x$ to the nearest integer.)
Is there some $\delta>0$ such that\[\lim_{x\to \infty}N(X,\delta)=\infty?\]
Graham proved this is true, and in fact\[N(X,1/10)> \frac{\log X}{10}.\]This was substantially improved by Sárközy
[Sa76], who proved that for, all sufficiently small $\delta>0$,\[N(X,\delta)>X^{1/2-\delta^{1/7}}.\]See also
[465] for upper bounds and
[953] for a similar problem.
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This page was last edited 16 September 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 #466, https://www.erdosproblems.com/466, accessed 2026-02-13
