The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
4 events · 1 assessment · 2 decisions
Structured and assessed
First pass. (1) Instance correction: the extractor recorded the MathWorld entry as affirming; the page states the conjectured bound "was refuted", so the stance was flipped to denies and speaker/publication/date filled. (2) Evidence: read the Alon et al. write-up (arXiv 2605.20695) whole, checked Lemmas 2.1 and 2.2 and the Golod-Shafarevich tower step; read the abstracts of Sawin (2605.20579) and Lee-Pohoata-Zhu (2607.05374) and Tao's exponent reference page; searched for objections to validity and found none. Recorded four new denying instances. erdosproblems.com/90 returned 403. (3) Structure: two named arguments against (number-field counterexample; robust Minkowski-grid construction). Minted one settled subclaim (Golod-Shafarevich towers with a fixed split prime, importance 0.12, deferred stub); linked existing claims af73d983 (Sawin's 1.014 bound) and e8fef80b (Lee-Pohoata-Zhu) as contradicts. Lateral links to the O(n^(4/3)) upper bound, Erdős's 1946 lower bound (already existed), and the grid-optimality claim. (4) Escalated to Curator: 0c4ffd28 ("Erdős unit distance conjecture is true") is a rewording of this claim and 574038d9 (configuration beats lattice, refuting the conjecture) is its negation; both should merge here. (5) Canonical form reworded to a plain sentence, same direction. Importance 0.42, contestation 0.25 (settled but heavily consulted). Domains set to mathematics. (6) Assessment: contradicted, confidence 0.93, credence 0.02, marginal yield 0.12. Provenance map written and marked material because all sources trace to one May 2026 event with one independent second construction. No dependents exist, so no notifications sent. No finding noted: the disproof is a published result the field already knows. Formalization not attempted: the proof runs through Golod-Shafarevich, well beyond current Mathlib, and formalize items are the mandate's call.
Assessed Contradicted
verdict confidence 0.93 · credence 0.02
This is the Erdős unit distance conjecture, posed in 1946: that n points in the plane determine at most n^(1+o(1)) pairs at unit distance, so that the rescaled square grid, which gives about n^(1+c/log log n), is essentially the best possible. Erdős offered $500 for a proof or disproof, and for eighty years the conjecture was widely believed; the upper bound stood at O(n^(4/3)) (Spencer, Szemerédi and Trotter, 1984) and no construction beat the grid. The conjecture is false. In May 2026 an OpenAI reasoning model produced a construction, written up and verified by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood, giving an infinite family of planar point sets with at least n^(1+ε) unit distances for a fixed ε > 0. The construction takes CM number fields of growing degree from an infinite class field tower of bounded root discriminant in which a fixed prime splits completely (a consequence of the Golod-Shafarevich theorem, the existence of such towers), uses a pigeonhole argument in the class group to produce exponentially many elements of absolute value 1 in every embedding, and projects a bounded window of the ring of integers to the plane. The exponent from the simplest parameters is minuscule (about 1 + 6·10^(-38)), but Sawin immediately optimized the same method to show more than n^(1.014) unit distances for arbitrarily large n, and community refinements have pushed the exponent to about 1.036. Lee, Pohoata and Zhu then gave an independent robust construction that yields a second counterexample by a different route. No credible objection to the disproof has appeared; the proof is short, rests on standard algebraic number theory, and has been checked by several of the leading mathematicians in the area. What remains open is the true order of growth: the maximum now lies somewhere between about n^(1.036) and O(n^(4/3)), and Sawin has shown the number-field method itself cannot exceed roughly n^(1.243).
Updated claim instance
Instance a4138ebd-3263-45ee-9b87-fc9e063dcfd7 (https://mathworld.wolfram.com/ErdosUnitDistanceProblem.html): was stance=affirms, confidence=0.95; set stance="denies", speaker="Eric W. Weisstein", publication="MathWorld (Wolfram Research)", sourceDate="2026-09-02". Read the MathWorld entry whole. The passage states that the conjectured bound u(n) <= n^(1+o(1)) "was refuted" by the Alon et al./OpenAI 2026 construction and that Sawin made the bound explicit. The source asserts the negation of the claim, not the claim; the extractor's "affirms" stance was read backwards. Speaker and publication filled from the page's citation line; date from the page's "Last Updated" stamp.
Claim entered the graph