Bhowmick 2024 problem erdos about rich distances

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Full paper in Markdown. The arXiv record (https://arxiv.org/abs/2407.01174, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Krishnendu Bhowmick, A problem of Erdős about rich distances. arXiv preprint (2024). arXiv:2407.01174, doi:10.48550/arXiv.2407.01174. Published as: A note on a problem of Erdős about rich distances. Studia Sci. Math. Hungar. 62 (2025), no. 1, 89--94, doi:10.1556/012.2025.04332. The copy read for this card is arXiv:2407.01174v3 (4 July 2024).

Bhowmick answers affirmatively an old question of Erdos on whether some set of n points can have c*n distances each occurring more than n times, exhibiting a set of n points in which floor(n/4) distances occur at least n+1 times (Theorem 1.1). A generalization gives, for each m, sets of n points in which at least floor(n/(2(m+1))) distances occur at least n+m times (Theorem 1.2; the abstract's "more than $n+m$ times", p. 1, is stronger than the theorem). The paper is six pages and construction-based; Theorems 1.1 and 1.2 give the two constructions. It was screened as a candidate source for problem 217 and confirmed unrelated: problem 217 concerns a set in which the i-th distance occurs exactly i times, a different configuration from richness above n.

For Problem 132, the construction is counterpressure rather than a solution: it shows that linearly many distance values can simultaneously have multiplicity greater than $n$, while Problem 132 asks how many occurring distances must have multiplicity at most $n$.

Source: https://arxiv.org/abs/2407.01174.

Bears on. #756 (Theorem 1.1 answers the problem's question affirmatively), #132 (counterpressure, as above), #217 (screened; does not apply)