Abstract
We study some discrete and continuous variants of the following problem of Erd\H os: given a finite subset $P$ of ${\Bbb R}^2$ or ${\Bbb R}^3$, what is the maximum number of pairs $(p_1,p_2)$ with $p_1,p_2\in P$ and $|p_1 -p_2|=1$?
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