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An old question posed by Erd{o}s asked whether there exists a set of n points such that c · 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 n/4 distances occur more than n times. We also present a generalized version, finding a set of n points where cₘ · n distances occurring more than $n+m$ times.
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Krishnendu Bhowmick (2024) studied this question.