A problem of Erdős about rich distances

2024/07/01 by Krishnendu Bhowmick, Bhowmick, Krishnendu
Computer Science · Mathematics · #52C10 #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2407.01174

openalex publication_date 2024/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An old question posed by Erdős 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 \lfloor (n)/(4)\rfloor distances occur more than n times. We also present a generalized version, finding a set of n points where c_m ⋅ n distances occurring more than n+m times.

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