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Let be a set of points. Must there be two distances which occur at least once but between at most pairs of points? Must the number of such distances as ?
Let be a set of points. Must there be two distances which occur at least once but between at most pairs of points? Must the number of such distances as ?
Source: erdosproblems.com/132
No claim settles this problem.
Open. The site labels the problem OPEN, and its commentary states the first question for , as the corrected Statement does.
The site's wording, as accessed 2026-09-04, places no bound on , and its first question fails for every . The smallest substantive failure is at : two unit equilateral triangles sharing an edge (a rhombus) give five pairs at distance and one pair at distance , so only the diameter occurs between at most pairs. For the failure is degenerate: one point determines no distance, two points one, and an equilateral triangle one. No failure is recorded for any , so these are boundary failures. The change inserts "" after "a set of "; nothing else changes, and the second question, which concerns large , is unaffected. The form is the poser's own. Erdős and Fishburn [ErFi95] (Section 5, pp. 145-146) note that the second diagram of their Fig. 1 (p. 143), this rhombus, has every distance below the diameter occurring more than times, attribute the conjecture to Erdős and Pach [ErPa90], and state it as their Conjecture 4 (p. 146): "There is no for such that for every interpoint distance less than ", which with the Hopf–Pannwitz bound on the diameter [HoPa34] is the first question for ; the library card records the paper. Erdős states it again with Pach in [Er97b] (item 11, p. 231), after Pannwitz's bound on the diameter: "can it happen that for every other distance occurs more than times? We believe that the answer is no!"; the library card records the item. The site's curator states the same form in the site's commentary under the label OPEN: "Erdős [Er84c] believed that for there must always exist at least two such distances. This is false for ", with the rhombus as witness. Clemen, Dumitrescu and Liu state it as Erdős's Conjecture 1.1 ([CDL25], arXiv:2505.04283v5, p. 2), "Let ", and add that "the condition is necessary" because of the rhombus; their theorems settle only special cases, so their statement is independent of any claim that would settle the corrected Statement. The copy of [Er84c] read for its library card (pp. 134-135) gives the Pannwitz bound on the diameter and questions on equal multiplicities but no statement of this question, so the site's and [CDL25]'s attribution to it is not confirmed there; [ErPa90] and [Er97e] are not held. The defect is the site's: the poser's statements read here carry the bound. The form was fixed from these sources before reading which results settle it. The counterexample at is recorded by the curator, by [CDL25] and in [ErFi95] itself; it settles no instance of the corrected Statement and counts for nothing. The problem's standing judges the corrected Statement.