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

Updated Oct 8

Problem 98

Statement. Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$ , with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$ ?

Status. Open: the site labels the problem OPEN (snapshot of 5 September 2026). No result about the problem has been claimed, so it has no claim page and the frontmatter standing is open.

Source. erdosproblems.com/98, snapshot accessed 2026-09-05. Cite as: T. F. Bloom, Erdős Problem #98, https://www.erdosproblems.com/98.

References.

[EFPR93] Erdős, Paul and Füredi, Zoltán and Pach, János and Ruzsa, Imre Z., The grid revisited. Discrete Math. (1993), 189--196.
[Du08] A. Dumitrescu, On distinct distances among points in general position and other related problems. Period. Math. Hungar. 57 (2008), 165--176, DOI 10.1007/s10998-008-8165-4.
[Ta24] T. Tao, Planar point sets with forbidden 4-point patterns and few distinct distances. arXiv:2409.01343v1; Discrete Comput. Geom. 76 (2026), 643--651, DOI 10.1007/s00454-025-00761-2.
[GGK25] A. Ghosal, R. Goenka, and P. Keevash, On subsets of lattice cubes avoiding affine and spherical degeneracies. arXiv:2509.06935v1; Discrete Comput. Geom. 76 (2026), 1886--1911, DOI 10.1007/s00454-026-00853-7.

Formalization. The statement is recorded in formal-conjectures.

Current assessment

The question (site formulation accessed). The statement above: whether the fewest distinct distances $h(n)$ determined by $n$ points in the plane with no three on a line and no four on a circle satisfies $h(n)/n\to\infty$ . The site reports that Erdős could not prove $h(n)\geq n$ and records the upper constructions of Pach and of Erdős, Füredi, Pach and Ruzsa [EFPR93]. The site labels the problem OPEN, its proof-claims tab carries no claim, and no forum claim, release manuscript or lead names the problem, so it has no claim page.

Known results. The constructions in Progress below give $h(n)=O(n^2/\sqrt{\log n})$ [Du08, Ta24]. No lower bound beyond the trivial one is compiled here, and an upper construction does not by itself certify that the problem is open.

Search scope. The site's problem page (snapshot of 2026-09-05) and the library cards of [Du08], [Ta24] and [GGK25]; no literature search beyond the site's references and those cards was made. The compiled results are statement and transfer records: no source proof was checked in full, and no acceptance review or formal verification is claimed.

Progress

The site reports that Erdős could not prove $h(n)\geq n$ , and reports upper constructions of Pach and of Erdős--Füredi--Pach--Ruzsa [EFPR93]; the EFPR paper is cited from the site's reference list, and its theorem is not compiled here.

Dumitrescu [Du08] defines general position to mean no three collinear and no four cocircular. Its Theorem 1 constructs $n$ points satisfying these conditions, and also avoiding parallelograms, with $O(n^2/\sqrt{\log n})$ distinct distances.

Tao [Ta24] gives a second upper construction aimed at the exact two E98 exclusions. Its Theorem 1.2 (arXiv v1, p. 2) supplies fixed $c>0,L_0$ and sets $A_L\subseteq\{0,\ldots,L-1\}^2$ with $|A_L|\geq cL$ for every sufficiently large $L$ . Remark 1.8 (p. 6) says those sets have no three collinear and no four concyclic, with the one-line reason that a non-degenerate parabola over $\mathbf F_p$ has these properties. That reason covers the collinearity half. The paper gives no further argument for the concyclicity half, which rests on the remark's assertion alone.

For an exact requested size $N$ , reduce $c$ to at most one, take $L=\lceil N/c\rceil$ , and retain any $N$ points of $A_L$ . Both exclusions survive taking subsets. The ambient grid has $O(L^2/\sqrt{\log L})=O(N^2/\sqrt{\log N})$ distances. This is an exact- $N$ upper construction and gives no new lower bound.

Ghosal, Goenka and Keevash [GGK25], Theorem 1.3 and Corollary 1.4 (arXiv v1, p. 3), give, for every sufficiently large $L$ , at least $7L/12$ points of the $L\times L$ grid with no four collinear or cocircular. Collinear triples remain allowed, so this is a nearby variant rather than an E98 construction.

The Tao and Ghosal--Goenka--Keevash statements above are those of the arXiv v1 records; both papers have since appeared in Discrete & Computational Geometry, as the references record.

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 (18)

Problem 217 Distance Problems distance_problems/chojecki_2026_erdos_problem_655_natural_repairs_exact Lemma 2.1 (p. 2): under the local condition A_m every point sees at least ceil((n-1)/m) distances distance_problems/dumitrescu_2008_distinct_distances_points_general_position Theorem 1: general-position, parallelogram-free sets with few distances Theorem 2 (p. 2): distinct-distance subsets on a line have order n^{1/2}distance_problems/erdos_1983_combinatorial_problems_geometry Added note (p. 54): h(n), the distinct distances forced by n points with no three on a line and no four on a circle distance_problems/erdos_1989_problem_leo_moser_about_repeated_distances Theorem 1 (p. 571): n points in general position in the plane with fewer than (3/2)n^{log 3/log 2} distinct distances distance_problems/ghosal_2025_subsets_lattice_cubes_avoiding_affine_spherical_degeneracies Corollary 1.4: at least 7n/12 grid points with no four collinear or concyclic distance_problems/sheffer_2014_distinct_distances_open_problems_current_bounds Problem 8 (p. 6): the asymptotic value of D_gen(n) for points in general position distance_problems/tao_2024_planar_point_sets_forbidden_4_point Remark 1.8: the constructed sets have no three collinear and no four concyclic points Theorem 1.2 (Main theorem): grid subsets of size order n avoiding all eight four-point patterns

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