The maximum number of unit distances among n planar points is O(n^(4/3)).
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
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.
Full reasoning: the evidence and decisions behind this verdict
The claim is a theorem with a refereed proof (Spencer, Szemerédi and Trotter, "Unit distances in the Euclidean plane", in Graph Theory and Combinatorics, Academic Press 1984, pp. 293–303) that has been independently re-derived by at least two different methods and is reproduced in standard texts (Pach and Agarwal, Combinatorial Geometry; Matoušek, Lectures on Discrete Geometry; Brass, Moser and Pach, Research Problems in Discrete Geometry). No source disputes it.
Two proofs were recorded as arguments and both go through. Székely's proof (Combinatorics, Probability and Computing 6, 1997) rests on the crossing lemma, that a graph with n vertices and m ≥ 4n edges has crossing number Ω(m^3/n^2), itself a two-line consequence of Euler's formula and a sampling argument; the unit-circle arcs form a multigraph with about u edges, edge multiplicity at most two, and at most n(n − 1) crossings because two unit circles meet in at most two points, so u^3/n^2 = O(n^2) and u = O(n^(4/3)). The original proof rests on the incidence bound O(n^(2/3) m^(2/3) + n + m) for n points and m unit circles, applied with m = n; each unit-distance pair is two point–circle incidences. Both premises are textbook results, and the second is also the two-degrees-of-freedom case of the Pach–Sharir incidence theorem for curves.
The instance set is uniform: every source read asserts the bound as an established theorem. These include the companion paper to the 2026 disproof by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood (arxiv.org/abs/2605.20695), which states the bound and names Székely's proof as the shortest; Guth's September 2026 survey (arxiv.org/abs/2609.10791), which states the bound "has not been improved"; Senger's note on an unfinished attempt to show 4/3 is not sharp (arxiv.org/abs/2605.26145); a 2024 paper on unit distances in arbitrary norms (arxiv.org/abs/2410.07557), which records that Székely's proof extends the bound to all strictly convex norms and that Valtr's norm attains it; and the MathWorld entry on the problem. The MathWorld passage is recorded with its formulas stripped in the stored text, so the mechanical quote check fails, but the page does state the bound; it attributes it loosely to a 2016 item by Szemerédi rather than to the 1984 paper.
An adversarial check found nothing that could lower the verdict: the 2026 lower-bound constructions (exponent about 1.014, refined to about 1.036) lie far below 4/3 and are consistent with the theorem; the only live questions are whether 4/3 can be improved (open; Katz and Silier have partial results in a related direction, and Sawin has shown the number-field construction cannot exceed about n^(1.243)) and what the sharp constant is. Neither bears on this claim's truth. The verdict is "verified" on the accepted-proof branch: the graph holds no machine-checked formal statement, and a Lean formalization of the O-bound over finite planar point sets would be routine but has not been done.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Draw the unit circle around each of the n points, discard circles carrying fewer than three points, and form the multigraph whose vertices are the points and whose edges are the arcs between consecutive points on each circle; this graph has about u edges, where u is the number of unit distances, at most two parallel copies of any edge, and at most 2·(n choose 2) = n(n − 1) crossings, since two unit circles meet in at most two points. Because a graph with n vertices and m ≥ 4n edges has crossing number Ω(m^3/n^2), and the bounded edge multiplicity costs only a constant factor, u^3/n^2 = O(n^2), which gives u = O(n^(4/3)).
The inference goes through. The only premise beyond elementary geometry is the crossing number inequality, a textbook theorem with a short proof from Euler's formula, and the remaining steps (two unit circles meet in at most twice, so the arc graph has at most n(n − 1) crossings; edge multiplicity is at most two) are checked in a line each. The argument also shows the bound for every strictly convex norm, since only the two-intersection property of the unit circles is used.
Centre a unit circle at each of the n points; a pair of points at unit distance is then a pair of point–circle incidences, so the number of unit distances is half the number of incidences between the n points and these n unit circles. Because the number of incidences between n points and m unit circles is O(n^(2/3) m^(2/3) + n + m), setting m = n gives O(n^(4/3)) incidences and hence O(n^(4/3)) unit distances.
The inference is immediate once the premise is granted: the argument stands or falls with the incidence bound for points and unit circles, which Spencer, Szemerédi and Trotter proved by cell decomposition in 1984 and which Clarkson, Edelsbrunner, Guibas, Sharir and Welzl reproved by cuttings in 1990; it is also the two-degrees-of-freedom case of the Pach–Sharir theorem. The reduction from unit distances to incidences with m = n circles is exact, each unit-distance pair contributing two incidences.
Provenance
Where this claim has been said, linked to its canonical form.
The best upper bound currently known is u(n)=O(n^(4/3)), where O is big-O notation (Szemerédi 2016).
Best known upper bound on unit distances.
Asserted without evidence of the source's own. The entry states the bound as standard background and cites a 2016 item by Szemerédi rather than the 1984 paper of Spencer, Szemerédi and Trotter in which it was proved; the attribution is loose but the statement itself is the standard one. The quoted passage was not found in the stored copy of this source.
U(n)\leq O(n^{4/3}). This was first proved by Spencer, Szemerédi and Trotter [32] in 1984. Several simpler proofs of the same bound up to a constant factor have been given over the years, the shortest and most elegant one is due to Székely [33].
Background section of the companion paper to the 2026 disproof of the Erdős unit distance conjecture, stating the best known upper bound and naming the original proof and Székely's simpler one.
In the 1980s, using an interesting argument based on topology, Spencer, Szemeredi, and Trotter [SST] proved that U_{max}(n)\leq Cn^{4/3}. This bound has not been improved.
Survey of the unit distance problem and incidence geometry written after the 2026 disproof of Erdős's conjecture; states the Spencer–Szemerédi–Trotter upper bound and that it has not been improved.
There is still a gap between the best known upper bound of $n^{\frac{4}{3}}$ due to Spencer, Szemerédi, and Trotter in [7], and the recent results.
Note describing an unfinished attempt to show the n^(4/3) bound is not sharp; states the bound as the best known upper bound.
Despite considerable effort, the best known upper bound on this problem is $O(n^{4/3})$ , proved in 1984 by Spencer, Szemerédi, and Trotter [8].
Introduction to a paper on unit distances in non-Euclidean norms; states the Euclidean upper bound and notes Székely's proof extends it to all strictly convex norms, where Valtr's norm shows it is tight.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.