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Claim. The first question of Problem 132 holds for $n=5$ and $n=6$: every set of five or six points in the plane determines two distinct distances each of which occurs between at most $n$ pairs. Paul Erdős and Peter C. Fishburn, Multiplicities of interpoint distances in finite planar sets, Discrete Appl. Math. 60 (1995), no. 1-3, 141-147, cited as [ErFi95] on the problem page; library home erdos_fishburn_1995_multiplicities_interpoint_distances_finite_planar_sets. The paper's Conjecture 4 (Section 5, pp. 145-146) is the first question in the form that some distance smaller than the diameter has multiplicity at most $n$, for every $n\ge5$; with the Hopf–Pannwitz bound [HoPa34], that the diameter itself occurs at most $n$ times, this gives the two required distances. For $n=5$, Theorem 2 (pp. 143-144) shows that a five-point set with two distances has multiplicities $(5,5)$, and a set with three or more distances has at most one multiplicity above $5$ because the ten pairs are shared among them. For $n=6$, Theorem 5 (p. 146) excludes every multiplicity vector $(a,14-a,1)$, and in particular $(7,7,1)$, by deleting an endpoint of the uniquely occurring distance and applying Altman's classification of convex two-distance pentagons (nonconvex five-point sets having more than two distances), which the paper uses to confirm the conjecture for six points. The paper leaves every $n\ge7$ open.

Covers. The first question for $n=5$ and $n=6$ only. The question for $n\ge7$ and the second, asymptotic question are not touched.

Depends on. No page of this wiki.

Acceptance. Refereed: the paper is the publisher's version of record in Discrete Applied Mathematics (the Crossref record dates the issue to June 1995 without a day, so this page is named by the first day of that month). Not reviewed under the corpus's rule: the site's commentary credits [ErFi95] with the cases $n=5$ and $n=6$, but the site labels the problem OPEN, so that commentary is a credit on an open problem and not an acceptance that settles it. Clemen, Dumitrescu and Liu restate the two cases as known in their 2025 paper. The proof was not independently reviewed by this corpus.