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

Updated Oct 8

Problem 89

Statement. Does every set of $n$ distinct points in $\mathbb{R}^2$ determine $\gg n/\sqrt{\log n}$ many distinct distances?

Status. Open. The site's export of 2026-09-04 labels the problem "OPEN" (page last edited 23 January 2026), and the site lists no proof claim for it.

Source. erdosproblems.com/89, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #89, https://www.erdosproblems.com/89.

References.

[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
[GuKa15] Guth, Larry and Katz, Nets Hawk, On the Erdős distinct distances problem in the plane. Ann. of Math. (2) (2015), 155-190.

Formalization. Statement in formal-conjectures.

Current assessment

Question and standing. The site formulation above (page last edited 23 January 2026) asks whether every $n$ -point planar set determines $\gg n/\sqrt{\log n}$ distinct distances. Nothing claims to settle it: the site lists no proof claim, no forum claim and no release item names the problem as its subject, and no literature result known here reaches the conjectured bound. The problem is open.
Known results. The $\sqrt n\times\sqrt n$ integer grid determines $O(n/\sqrt{\log n})$ distinct distances, so the bound asked for would be best possible. Guth and Katz [GuKa15], carded at guth_2015_erdos_distinct_distance_problem_plane, proved that every $n$ -point planar set determines $\gg n/\log n$ distinct distances, which leaves a factor of $\sqrt{\log n}$ . The site records stronger forms, that a single point determines $\gg n/\sqrt{\log n}$ distinct distances, that $\gg n$ points do, or Erdős's 1975 conjecture [Er75f], printed in Section 1 of the survey carded at erdos_1975_problems_elementary_combinatorial_geometry, that the counts of distinct distances from the points sum to $\gg n^2/\sqrt{\log n}$ , under Problem 604; the related Problem 661; and the generalization to higher dimensions under Problem 1083.
A release item that claims nothing here. The OpenAI Math Release's preprint The Falconer distance conjecture in all dimensions (23 September 2026), with Lean in the release's formalization, states that a compact set in $\mathbb R^d$ , $d\ge2$ , of Hausdorff dimension greater than $d/2$ has a distance set of positive Lebesgue measure. That is the continuum analogue of the distinct-distances questions and says nothing about the number of distances of a finite planar set; for finite point sets the manuscript points to the release's distinct-distances theorem in dimensions $d\ge3$ , recorded under Problem 1083. That theorem's manuscript reproves, in its Appendix B, the planar Guth--Katz bound $\gg n/\log n$ recorded above (Theorem B.12), without improving it, and has no claim page here. The item is background here and gives no claim page.
Status search. The site's page and its proof-claims tab,; no broader literature search is recorded.
Proof coverage and review. None: the problem is open, and no proof is held or reviewed.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (25)

Problem 604 Distance Problems Distance Problems distance_problems/clarkson_1990_combinatorial_complexity_bounds_arrangements_curves_spheres Corollary 5.6 (p. 136): g(m) = Omega(m^{7/4}), so some point of every planar m-set has Omega(m^{3/4}) distinct distances distance_problems/erdos_1946_sets_distances_points Theorem 1 (p. 248): n points in the plane determine between (n - 3/4)^{1/2} - 1/2 and cn/(log n)^{1/2} distinct distances at the minimum distance_problems/erdos_1975_problems_elementary_combinatorial_geometry Section 1, inequality (1) (p. 99): bounds for the fewest distinct distances among n planar points distance_problems/erdos_1983_combinatorial_problems_geometry Distinct distances (p. 53): Erdős's conjecture that n points determine about n/sqrt(log n) distances distance_problems/erdos_1985_problems_results_combinatorial_geometry Conjecture, p. 2, display (4): f_2(n) > cn/(log n)^{1/2} and P_2(n) < n^{1+c/log log n}On the Erdős distinct distance problem in the plane Theorem 1.1 (p. 155): N points in the plane determine ≳ N/log N distinct distances distance_problems/katz_2004_new_entropy_inequality_erdos_distance_problem Corollary 6 (p. 6): a point with Omega(n^{(48-14e)/(55-16e)-eps}) distinct distances distance_problems/openai_2026_higher_dimensional_erdos_distinct_distances_conjecture Theorem B.12: n points in the plane determine at least c n / log(2n) distinct distances distance_problems/pach_2002_isosceles_triangles_determined_planar_point_set Theorem 1: n planar points span O(n^{(11e-3)/(5e-1)+eps}) isosceles triangles Theorem 2: incidences between n points and l circles with m distinct centers distance_problems/sheffer_2014_distinct_distances_open_problems_current_bounds Problem 1 (p. 1): the exact asymptotic value of D(n), between Omega(n/log n) and O(n/sqrt(log n))integer_sequences/erdos_1997_some_old_new_problems_various_branches_combinatorics

Graph