A problem of Erdős about rich distances

Abstract

An old question posed by Erdős asked whether there exists a set of $n$ points such that $c\cdot n$ distances occur more than $n$ times. We provide an affirmative answer to this question, showing that there exists a set of $n$ points such that $\lfloor \frac{n}{4}\rfloor$ distances occur more than $n$ times. We also present a generalized version, finding a set of $n$ points where $c_m\cdot n$ distances occurring more than $n+m$ times.

6 pages, 4 figures