The maximum number of unit distances among n planar points is at least n^(1+c/log log n) for some constant c > 0.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
This is Erdős's 1946 lower bound for the unit distance problem, and it is a theorem with a short, elementary proof that has stood for eighty years without objection. Place n points on a square grid of side about √n and choose an integer m of order n with unusually many representations as a sum of two squares; every interior grid point then lies at squared distance m from many other grid points, and rescaling so that √m becomes the unit produces the claimed number of unit distances. The only ingredient beyond counting is the maximal order of the two-squares representation function, a standard consequence of its product formula together with the distribution of primes congruent to 1 modulo 4.
The bound was long conjectured to be essentially sharp. That conjecture was disproved in 2026, when a number-field construction gave more than n^(1.014) unit distances for arbitrarily large n. The stronger lower bound supersedes this one as the state of the art but does not bear on its truth; the 1946 bound remains correct and is universally cited as the historical starting point of the problem.
Full reasoning: the evidence and decisions behind this verdict
The proof was checked directly rather than taken on authority. Let the point set be the grid {0, ..., k-1}^2 with n = k^2. Choose m at most k^2/4 with all prime factors congruent to 1 mod 4, so that m = a^2 + b^2 has r_2(m) ordered integer solutions, all with |a|, |b| ≤ k/2. Every grid point in the central quarter of the grid has r_2(m) grid neighbours at squared distance m, so the grid, rescaled by 1/√m, has at least (n/4) · r_2(m) / 2 unit-distance pairs. Taking m to be the product of the first j primes congruent to 1 mod 4 gives r_2(m) = 4 · 2^j; by the prime number theorem in arithmetic progressions the j-th such prime is about 2j log j, so log m is about j log j and j is about log m / log log m, whence r_2(m) ≥ m^((log 2 - o(1)) / log log m). With m of order n this yields u(n) ≥ n^(1 + c/log log n) for any fixed c below log 2 and large n, and adjusting the constant covers small n. The only non-trivial ingredient is the number-theoretic lemma recorded as the maximal order of the two-squares representation function, a standard consequence of the product formula for r_2 and found in Hardy and Wright.
Three recorded instances affirm the bound with attribution to Erdős's 1946 paper, and none denies it: the MathWorld entry (mathworld.wolfram.com/ErdosUnitDistanceProblem.html), the collective introduction of the 2026 nine-author note on the disproof of the unit distance conjecture, and Noga Alon's section of the same note (arxiv.org/abs/2605.20695, full text at arxiv.org/html/2605.20695v1), which states "Erdős proved that U(n) ≥ n^{1+Ω(1/log log n)}". The two arXiv passages are one document under two addresses, so the instance set is two independent voices restating one primary source. None adds argument; the primary source (Erdős, American Mathematical Monthly 53 (1946), 248–250, doi.org/10.2307/2305092) is short and its argument is the one reproduced above and in Brass, Moser and Pach, Research Problems in Discrete Geometry, chapter 5. The recorded passages could not be matched mechanically against the stored page texts because both MathWorld and the arXiv HTML render formulas in ways that duplicate or replace the symbols; the surrounding sentences were read directly and say what the instances record, so all three stand.
The May 2026 developments (the OpenAI construction, the explicit n^(1.014) bound of arxiv.org/abs/2605.20579, and the robust version of Lee, Pohoata and Zhu) supersede this bound but do not bear on its truth; a stronger lower bound is consistent with a weaker one. What would change the verdict: an error in the maximal order of the two-squares function, which is excluded by the explicit product formula; nothing else in the argument is delicate. The status is "verified" in the accepted-proof sense of the mathematics domain, not machine-checked; no formal statement has been published for this claim and its importance does not call for one.
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.
Place the n points on a square integer grid of side about √n and choose an integer m at most about n that, by the maximal order of the two-squares representation function, has at least m^(c/log log m) representations as a sum of two squares; since every grid point lies at squared distance m from roughly r_2(m) other grid points (a constant fraction of them inside the grid), rescaling the grid so that √m becomes the unit yields n · n^(c'/log log n) unit distances, which is the claimed bound.
The inference goes through: given an integer m of order n with m^(c/log log m) representations as a sum of two squares, the grid count is a direct calculation, and the loss from grid points near the boundary only changes the constant. The argument rests entirely on the maximal order of the two-squares representation function, an elementary consequence of the product formula for that function together with the distribution of primes congruent to 1 modulo 4, and nothing in the discourse questions it.
Provenance
Where this claim has been said, linked to its canonical form.
Erdős (1946) proved a lower bound of u(n)=n^(1+Omega(1/lnlnn)), where Omega is big-Omega notation
Erdős's original lower bound for the unit distance problem.
Asserted without evidence of the source's own. An encyclopedia entry that reports the 1946 lower bound with attribution to Erdős's paper and offers no proof or derivation of its own; its account of the bound is a restatement of the primary source. The quoted passage was not found in the stored copy of this source.
In his original work, he noted that a set of n points may have at most O(n^{3/2}) unit distances via noting that the unit distance graph cannot contain a K_{2,3} (two unit circles can only intersect in at most 2 points) while using a √n × √n grid to show that a set of n points may have n^{1+Ω(1/log log n)} unit distances.
The collective introduction (Section 1.1, History of the problem) of the nine-author expository note on the 2026 disproof of the unit distance conjecture, summarizing Erdős's 1946 paper.
Asserted without evidence of the source's own. The introduction restates the bound and names the grid construction as its source, but proves nothing about it; the paper's own mathematics concerns the number-field construction that beats this bound. The quoted passage was not found in the stored copy of this source.
Erdős proved that $U(n) \geq n^{1+\Omega(1/\log\log n)}$ , and the best known upper bound is $U(n) \leq O(n^{4/3})$ .
Alon's contributed section summarizing the history of the unit distance problem before discussing the 2026 disproof of the conjectured matching upper bound.
Asserted without evidence of the source's own. Alon's remarks restate the 1946 lower bound with attribution to Erdős's paper and offer no derivation; the section is a historical framing, not an argument for the bound. The passage sits in the full text of the paper rather than on the abstract page under which the instance was recorded. The quoted passage was not found in the stored copy of this source.
How these sources relate
- https://mathworld.wolfram.com/ErdosUnitDistanceProblem.html restates On sets of distances of n points (Erdős, American Mathematical Monthly 53, 1946), faithfully. The entry attributes the bound directly to Erdős's 1946 Monthly paper and restates it without adding any argument; the paper's lattice construction does give exactly this bound, so the restatement is faithful. Judged from the citing document alone.
- https://arxiv.org/abs/2605.20695 restates On sets of distances of n points (Erdős, American Mathematical Monthly 53, 1946), faithfully. Alon attributes the bound to Erdős's 1946 paper and restates it without new argument; the paper's lattice construction gives exactly this bound, so the restatement is faithful. Judged from the citing document alone.
- https://arxiv.org/html/2605.20695v1 restates On sets of distances of n points (Erdős, American Mathematical Monthly 53, 1946), faithfully. The introduction attributes the bound and the grid method to Erdős's 1946 paper and adds no argument; the attribution and the stated bound match what the paper proves. Judged from the citing document alone.
- https://arxiv.org/html/2605.20695v1 republishes https://arxiv.org/abs/2605.20695. The two URLs are the abstract page and the experimental HTML full text of the same arXiv submission 2605.20695v1, same authors and title, as shown by the arXiv identifier on both pages.
Assessment history
0 status changes over 2 assessments. full history →
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.