Problem 97

../

claims/: The 1 claim page of Problem 97, one per claimant's result; the problem's standing derives from them.

---

Statement. Does every convex polygon have a vertex with no other $4$ vertices equidistant from it?

Formulation. Erdős first asked the question with three in place of four [Er46b, p. 248]: does every convex polygon have a vertex from which no three vertices of the polygon are equally distant? He offered it as a strengthening of his conjecture that the vertices of a convex $n$-gon determine at least $[n/2]$ distinct distances, since such a vertex has $[n/2]$ distinct distances to the others. Danzer disproved it with a convex nonagon of threefold rotational symmetry, built from a Reuleaux triangle, in which every vertex has three other vertices at a common distance from it; Erdős reports the example and then asks the four-vertex question, which is the Statement [Er87b, pp. 175–176, Fig. 5]. The site states the problem with four, and that question sets the standing. The general form, in which some fixed $k$ replaces four, is discussed with the claims under Current assessment.

Status. Falsifiable on the site: its export of 2026-09-04 records the label "FALSIFIABLE", and its page was last edited 27 October 2025; its proof-claims tab carries one full proof claim, the manuscript of Liam Kruer, Jensen Kohlmeyer and Liam Price of 13 September 2026, recorded on its claim page without being adopted; the standing in the frontmatter is derived from the claim pages.

Source. erdosproblems.com/97, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #97, https://www.erdosproblems.com/97.

References.

• [Er46b] Erdős, P., On sets of distances of $n$ points. Amer. Math. Monthly (1946), 248-250.
• [Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
• [Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.
• [FiRe92] Fishburn, P. C. and Reeds, J. A., Unit distances between vertices of a convex polygon. Comput. Geom. (1992), 81-91.

Formalization. Statement in formal-conjectures.

Current assessment

Question and standing. The site formulation above asks whether every convex polygon has a vertex from which no four other vertices are equidistant, the distance allowed to depend on the vertex. The site labels it falsifiable: one convex polygon in which every vertex has four other vertices at a common distance from it would refute it, a finite check once the polygon is given, and that is a note on an open question, not a claim. One pending full claim asserts exactly such polygons: Kruer, Kohlmeyer and Price's manuscript constructs, for every large $n$, $n$ points in strictly convex position each with at least $(\frac14-o(1))\log_2\log_2n$ others at distance exactly one, with a counterexample of at most $3432\cdot2^{36036}$ vertices, and refutes as well the general form in which some fixed $k$ replaces four. The manuscript is unrefereed and unreviewed, this corpus has not built its Lean file, and the site's export of 2026-09-04 labels the problem "FALSIFIABLE" with no acceptance recorded, so the claim is pending and the frontmatter standing is claimed through it, not solved.

The site's remarks tie this problem to Problem 96: a positive answer here for $k+1$ equidistant vertices gives, by induction, at most $kn$ unit distances among $n$ points in convex position. Conversely, the accepted disproof of Problem 96 by Kruer and Kohlmeyer on [[problems/distance_problems/E0096/claims/2026_09_10_kruer_kohlmeyer|its claim page]] yields a counterexample here: their construction with $d=13$ has more than three unit pairs per point, and deleting points with at most three others at distance one leaves a nonempty set in strictly convex position in which every point has at least four, as that page records; Kruer, Kohlmeyer and Price state that strictly convex sets with arbitrarily large minimum unit-distance degree exist exactly when the ratio of unit pairs to points is unbounded over strictly convex sets (Lemma 4.1 and Remark 4.5 of their manuscript); since this problem lets the common distance vary with the vertex, only the direction from Problem 96 to this problem follows. Kruer and Kohlmeyer's explanation does not state this consequence, so no claim page attributes it to them, and the standing of this problem is derived from its own claim pages.

The three-neighbor example below is distinct from the four-neighbor question and supplies no proof or disproof of this catalog statement. Small cases are excluded in the problem's discussion thread: a post of 14 September 2026 by the account veljjanoski reports that no counterexample has at most ten vertices, with Nullstellensatz certificates for at most nine vertices and Gröbner-basis computations for ten, in code the account published, and the account sallerk reported on 17 September 2026 that it had rechecked the certificates for seven to nine vertices; a post of 15 June 2026 by the account mysticflounder reports a Lean exclusion of nine-point counterexamples in a repository that was no longer public on 7 October 2026. These are thread posts with code and no manuscript, so they have no claim page. The site's remark that Erdős in 1975 [Er75f] credited Danzer with a disproof of the general form for every $k$, a claim he did not repeat and the site presumes mistaken, names no manuscript and has no claim page either.

The account mysticflounder announced a proof on 18 May 2026: a reduction of the problem to nine-point sets, resting on a descent step that removes a vertex from any larger counterexample, which a reader questioned on 20 May. Its post of 15 June 2026 softened that announcement to significant progress toward a proof and listed the descent step as two open conjectures. Its post of 14 July 2026 reported the proof closed apart from a shared-radius pair residual, and a later edit qualified that report and marked the linked gist out of date, as retained in the complete post 7604. None of these posts supplies a completed proof. The record is the linked transcription of post 7604 (account mysticflounder, 14 July 2026); no accepted resolution follows. See the source provenance and limits.

Known Results

Er87b, printed pp. 175–176, Fig. 5, reports Danzer's convex nonagon with three equidistant neighbors at every vertex, and asks the four-neighbor question. The source prints three distance relations and a geometric construction but no numerical coordinates.

The sallerk forum source gives six coordinates. Its selected exact completion is a nine-point strictly convex set whose maximum distance multiplicity is exactly three at every vertex, with independently reviewed exact evidence. It realizes the Er87b relations; it is not identified as Danzer's original coordinate choice. All 36 distances and all 63 supporting-edge signs are checked, not only the advertised triples.

The separate uniqueness, degree-four, mirror-exclusion and minimum-size reports are qualified at that owner. In particular the claimed lower bound $n_3\geq7$ depends on an uninspected six-point exclusion; it is not a consequence of the nine-point witness. No variant inherits a newly proved status from this account.

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.

• erdos_1987_combinatorial_metric_problems_geometry
• erdos_1987_combinatorial_metric_problems_geometry / conjecture_p176
• erdos_1946_sets_distances_points
• erdos_1946_sets_distances_points / conjecture_p248
• erdos_1975_problems_elementary_combinatorial_geometry
• erdos_1975_problems_elementary_combinatorial_geometry / section_1_convex_polygons_p100
• erdos_fishburn_1995_multiplicities_interpoint_distances_finite_planar_sets
• furedi_1990_maximum_number_unit_distances_convex_n_gon
• furedi_1990_maximum_number_unit_distances_convex_n_gon / remark_1_2
• mysticflounder_2026_shared_radius_residual
• mysticflounder_2026_shared_radius_residual / mysticflounder_2026_shared_radius_residual
• sallerk_2026_convex_nonagon_relations
• sallerk_2026_convex_nonagon_relations / nonagon_from_relations