Problem 92

../

claims/: The 1 claim page of Problem 92, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb{R}^2$ in which every $x\in A$ has at least $f(n)$ points in $A$ equidistant from $x$.

Is it true that $f(n)\leq n^{o(1)}$? Or even $f(n) < n^{O(1/\log\log n)}$?

Status. Disproved. The site's export of 2026-09-04 labels the problem "DISPROVED" (page last edited 21 May 2026), and its remarks say that the disproof of Problem 90 disproves this stronger form; see the claim page.

Source. erdosproblems.com/92, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #92, https://www.erdosproblems.com/92.

References.

• [ErFi97] Erdős, Paul and Fishburn, Peter, Minimum planar sets with maximum equidistance counts. Comput. Geom. (1997), 207-218.
• [JJMT24] B. Janzer, O. Janzer, A. Methuku, and G. Tardos, Tight bounds for intersection-reverse sequences, edge-ordered graphs and applications. arXiv:2411.07188 (2024).
• [PaSh92] Pach, János and Sharir, Micha, Repeated angles in the plane and related problems. J. Combin. Theory Ser. A (1992), 12-22.
• [OpenAI26] OpenAI, Planar Point Sets with Many Unit Distances. Unnumbered 18-page technical report (2026).
• [ABGLSSTWW26] N. Alon, T. F. Bloom, W. T. Gowers, D. Litt, W. Sawin, A. Shankar, J. Tsimerman, V. Wang, and M. Matchett Wood, Remarks on the disproof of the unit distance conjecture, arXiv:2605.20695v1 (2026).

Formalization. Statement in formal-conjectures.

Current assessment

Disproved. The site formulation above (page last edited 21 May 2026) asks whether $f(n)\le n^{o(1)}$, or even $f(n)<n^{O(1/\log\log n)}$. The answer to both is no: one accepted claim page records OpenAI's report Planar Point Sets with Many Unit Distances (20 May 2026), whose Theorem 1.1 gives $N$-point planar sets with at least $N^{1+\delta}$ unit-distance pairs and whose text notes that pruning the unit-distance graph to minimum degree $N^{\Omega(1)}$ refutes the Erdős--Fishburn bound $k\le n^{o(1)}$; the acceptance evidence is the check of the theorem that Daniel Litt documents in the companion manuscript and the curator's label, and the standing derives from it. The companion manuscript of Alon and coauthors and Sawin's explicit construction, both claims on Problem 90, prove fixed-power unit-distance sets without stating this problem's consequence, so they have no claim page here; the transfer from the companion construction recorded below is the corpus's own deduction.

The lower bounds below contradict both proposed scales. The two fixed-power proof chains and the transfer below are recorded at their declared dependency boundaries. The original branch's independent review, including its E92 transfer, is retained as the full review; the companion chain and its minimum-degree transfer are author-recorded. The linked formalization records a statement and is not evidence of a checked formal proof.

The OpenAI report's own statements about its AI authorship and about later AI-assisted verification and review by external mathematicians are historical attestations rather than publication, acceptance, or formal-verification evidence; the acceptance evidence is the check Daniel Litt documents in Section 6 of the companion manuscript, beside the curator's label, as the claim page records. The independent corpus reviews do not recursively prove the named outside theorems, and no Lean build or proof is claimed. No dated status-search scope is recorded on this page.

Progress

The original OpenAI fixed-power theorem and the human companion's Theorem 1.1 each give an unbounded sequence of planar point sets whose unit-distance graphs have at least $N^{1+\eta}$ edges for one fixed $\eta>0$. The original branch's proof chain and its E92 transfer passed independent review relative to seven declared external premises and two exact shared companion-lemma scopes, retained as the full review. The companion chain, relative to six declared external inputs, and the minimum-degree transfer from the companion construction to this problem are author-recorded.

The original report uses an everywhere-unramified pro-$3$ tower, many fixed split rational primes, and exponent one. The companion uses a pro-$2$ tower, the single fixed split prime $101$, and one large common exponent. The two records share the norm-one and lattice-window mechanism but preserve their different arithmetic constructions.

Known Results

Let $G$ be the unit-distance graph of either fixed-power construction, the OpenAI report's or the companion's, on $N$ vertices, with at least $N^{1+\eta}$ edges. Repeatedly delete a vertex whose current degree is less than $N^\eta$. The process cannot delete every vertex: if it did, each original edge would be counted exactly once when its first endpoint was removed, giving strictly fewer than $N\cdot N^\eta=N^{1+\eta}$ removed edges.

A nonempty induced subgraph on $m$ vertices therefore remains with minimum degree at least $N^\eta$. Consequently

$$m\geq N^\eta+1, \qquad f(m)\geq N^\eta\geq m^\eta.$$

The first inequality makes these values of $m$ unbounded. Every counted neighbor is at the common distance one from its vertex, so the displayed fixed-power lower bound contradicts both $f(n)\leq n^{o(1)}$ and the proposed $n^{O(1/\log\log n)}$ scale.

The full original arithmetic and geometric chain is filed under [[../library/discrete_geometry/openai_2026_planar_point_sets_many_unit_distances/_index|the OpenAI report]]. The complete human companion chain is filed under [[../library/discrete_geometry/alon_2026_remarks_disproof_unit_distance_conjecture/_index|Alon et al. (2026)]]. The companion's Proposition 2.3 is a proof pointer relative to Hajir--Maire--Ramakrishna and Chebotarev and is not used in its Theorem 1.1.

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.

• alon_2026_remarks_disproof_unit_distance_conjecture
• alon_2026_remarks_disproof_unit_distance_conjecture / theorem_1_1_e90_e92
• openai_2026_planar_point_sets_many_unit_distances
• openai_2026_planar_point_sets_many_unit_distances / evidence/verify/full_review
• openai_2026_planar_point_sets_many_unit_distances / theorem_1_1
• janzer_2024_tight_bounds_intersection_reverse_sequences_edge
• janzer_2024_tight_bounds_intersection_reverse_sequences_edge / corollary_1_12