The maximum number of unit distances among n planar points is O(n^(4/3)).
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. The claim is the Spencer–Szemerédi–Trotter theorem (1984), a settled result; assessed verified (accepted-proof branch, no machine-checked statement) with credence 0.995, confidence 0.97, marginal yield 0.05. Structure: two named "for" arguments recording the two proofs the discourse treats as distinct, each with one load-bearing premise minted as a new subclaim after match_claim reported novelty: the crossing lemma (stated in Ω-form rather than with the 1/64 constant, since the Matcher noted constants would be identity-bearing and the discourse names the lemma by its order) and the point–unit-circle incidence bound. Both were scored 0.15 importance / 0.03 contestation as textbook bedrock and left as deferred stubs, with seed credence 0.99 each. Clarkson et al.'s cutting proof is a reproof of the incidence premise rather than a distinct route to the unit-distance bound, so it lives in the second argument's evaluation rather than as a third argument. Lateral structure: existing related links to the lower-bound claims and the contradicted Erdős conjecture were already in place and are apt. Proposed a supports edge from this claim into "the exact asymptotic order ... is unknown" (32d34edd), whose Steward decides. Did not mint a claim for "the exponent 4/3 can be improved"; it is live in the discourse (Senger, Katz–Silier) but no source states it cleanly enough yet and the Extractor will surface it if a paper does. Instances: four new affirming instances recorded from sources read during the web search (Alon et al. 2026 companion paper, Guth 2026 survey, Senger 2026, the 2024 arbitrary-norms paper). MathWorld instance read in full; reading recorded as asserts-without-evidence; its quote check fails only because formulas are stripped in the stored text. Source map written as immaterial. Importance set to 0.22 (contestation 0.15): settled theorem, slightly above the 0.15 anchor because it is the standing upper bound in a problem made very live by the May 2026 disproof of the Erdős conjecture. Canonical form left unchanged; it is already the neutral fifteen-word statement. No formal statement published: a Lean rendering would be routine but is a mandate-level formalize decision for a claim of this importance. No dependents exist to notify; the parent edge is proposed, not written.
Assessed Verified
verdict confidence 0.97 · credence 0.99
This is the Spencer–Szemerédi–Trotter theorem, proved in 1984: any n points in the plane determine at most a constant times n^(4/3) pairs at unit distance. It is a settled theorem of incidence geometry with several independent published proofs. The original argument treats unit distances as incidences between the points and the unit circles centred at them and proves a Szemerédi–Trotter-type incidence bound for points and unit circles; Clarkson, Edelsbrunner, Guibas, Sharir and Welzl reproved that bound in 1990 by random-sampling cuttings; and Székely's 1997 proof derives the unit-distance bound in a few lines from the crossing number inequality for graphs. The constant has since been made explicit, and Ágoston and Pálvölgyi (2022) showed the number of unit distances is below 1.94 n^(4/3). The bound has stood for four decades as the best known upper limit, and it remains so after the May 2026 disproof of Erdős's conjecture that the count is at most n^(1+o(1)). That disproof, which produced point sets with more than n^(1.014) unit distances, does not touch this theorem: the true growth rate now lies somewhere between about n^(1.03) and n^(4/3). Whether the exponent 4/3 can be lowered is a separate, open question. Székely's proof shows the same bound holds for every strictly convex norm on the plane, and Valtr exhibited a strictly convex norm for which n^(4/3) is attained, so any improvement must use a property special to the Euclidean distance.
Claim entered the graph