Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 96

claims/: The 3 claim pages of Problem 96, one per claimant's result; the problem's standing derives from them.

Statement. If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart.

Status. Open on the site: its export of 2026-09-04 records the label "OPEN", and its page, last edited 23 January 2026, carries no proof claim and no proof exposition. The standing in the frontmatter derives from the claim pages: the disproof of Kruer and Kohlmeyer, a Lean proof that the bounty site Conjectures.io verified on 10 September 2026 and certified, is accepted on that certification alone, with no refereed publication and no acceptance by erdosproblems.com, on its claim page; the stronger disproof of Kruer, Kohlmeyer and Price, a manuscript with a Lean file of 13 September 2026 giving $\Omega(n\log\log n)$ unit distances and posted as a proof claim under Problem 97, is pending on its claim page; Khopkar's 2016 preprint claiming the linear bound is rejected on its claim page. The disproof answers the Statement: the maximum number of unit-distance pairs among $n$ points in strictly convex position is not $O(n)$ . The site's remarks record that a positive answer here would follow from a positive answer to Problem 97; in the other direction, Kruer and Kohlmeyer's construction yields a counterexample to Problem 97 by deleting points of low unit degree, as their claim page records.

Source. erdosproblems.com/96, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #96, https://www.erdosproblems.com/96.

References.

[Ag15] Aggarwal, Amol, On unit distances in a convex polygon. Discrete Math. (2015), 88-92.
[BrPa01] Brass , Peter and Pach, János, The maximum number of times the same distance can occur among the vertices of a convex $n$ -gon is $O(n\log n)$ . J. Combin. Theory Ser. A (2001), 178-179.
[EdHa91] Edelsbrunner, Herbert and Hajnal, Péter, A lower bound on the number of unit distances between the vertices of a convex polygon. J. Combin. Theory Ser. A (1991), 312-316.
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
[Fu90] Füredi, Zoltán, The maximum number of unit distances in a convex $n$ -gon. J. Combin. Theory Ser. A 55 (1990), 316-320.

Formalization. Statement in formal-conjectures, which at the linked commit of 2026-09-18 tags erdos_96 research open with its answer unfilled. The accepted Lean proof establishes the negation of that statement with its answer fixed to true; its formal target is on the [[problems/distance_problems/E0096/claims/2026_09_10_kruer_kohlmeyer|Kruer–Kohlmeyer claim page]]. The pending manuscript's Lean file, which reproduces the problem definitions itself, is linked on the [[problems/distance_problems/E0096/claims/2026_09_13_kruer_kohlmeyer_price|Kruer–Kohlmeyer–Price claim page]]. This corpus has built neither.

Progress

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Known Results

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Linked library material

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Linked from (20)

Khopkar's claimed linear bound Superlinearly many unit distances in convex position Kruer, Kohlmeyer and Price's convex sets with n log log n unit distances problems/distance_problems/E0096/claims Problem 97 Kruer, Kohlmeyer and Price's convex sets with many unit neighbors Distance Problems distance_problems/aggarwal_2015_unit_distances_convex_polygon Theorem 1 (p. 3): U_c(n) <= n log_2 n + 4n unit distances among the vertices of a convex n-gon Theorem 2 (p. 3): no cycle with an intersection-free edge is a distance-like matrix Theorem 3 (p. 4): a 2^m x 2^m distance-like matrix with 2^{m-1}(m+1) entries equal to 1 distance_problems/edelsbrunner_1991_lower_bound_number_unit_distances_convex_polygon Lemma (p. 314, unnumbered): 0 < |B, b_1| < |A, a_1| if |A, a_1| is sufficiently small Theorem (p. 312, unnumbered): f(n) >= 2n - 7 unit distances among the vertices of a convex n-gon Erdős–Fishburn: Multiplicities of interpoint distances in finite planar sets Theorem 3: known bounds on the largest multiplicity of a distance in a convex n-gon distance_problems/furedi_1990_maximum_number_unit_distances_convex_n_gon Corollary 2.3 (p. 319): at most (a+b)(2 log_2(a+b) - 1) unit distances across a line in a convex set Lemma 2.1 (p. 317): a 0-1 matrix avoiding a 2-by-3 pattern has at most a + (a+b) floor(log_2 b) ones Theorem 1.1 (p. 316): f(n) < c n log n unit distances among the vertices of a convex n-gon

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