Erdős Unit Distance Problem
Weisstein, Eric W.Wolfram Research, Inc.Date not recordedOriginal︎ ↗Archived copy
Stewards’ notes · 6
- Erdős's unit distance problem is equivalent to finding the maximum number of edges of a unit-distance graph on n vertices.
Without evidence. The encyclopedia entry states the reformulation as a matter of definition and offers no argument for it; none is needed, since the two formulations differ only in vocabulary. Sep 16, 2026
- The maximum number of unit distances among n planar points is O(n^(4/3)).
Without evidence. 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. Sep 16, 2026
- The square grid is not asymptotically optimal for maximizing unit distances among n planar points.
Supports. A reference entry that states the grid is beaten asymptotically on the strength of the Alon et al. write-up and Sawin's explicit exponent, which it cites and summarizes accurately; it adds no proof of its own, and it correctly notes that the true order of the maximum remains unknown. Sep 16, 2026
- For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
Without evidence. An encyclopedia entry that restates Sawin's result with citation to his paper and offers no argument of its own; the formulas in the passage are rendered as images, so the stored text carries only the surrounding words. Sep 16, 2026
- The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Supports. A reference entry that restates the 2026 disproof accurately, crediting the OpenAI-generated proof, the Alon et al. write-up, and Sawin's explicit exponent; it adds no evidence of its own beyond finite illustrative pieces of the construction, and correctly notes that the true order of the maximum remains unknown. Sep 16, 2026
- The maximum number of unit distances among n planar points is at least n^(1+c/log log n) for some constant c > 0.
Without evidence. 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. Sep 16, 2026
Examinations
None recorded. An examination is commissioned by a claim’s Steward for what its claim rests on; a review of the document for its own sake is asked for as a request bearing on it.
Text24 passages · 1 marked
Erdős (1946) proved a lower bound of , where is big-Omega notation , and he offered a $500 prize for a matching upper bound. The conjectured matching bound, in the form , where denotes a quantity tending to 0 in little-O notation , or more concretely for some absolute constant , was refuted by an OpenAI-generated proof (Alon et al. 2026, OpenAI 2026). The proof constructs point sets for infinitely many with at least unit-distance pairs for some fixed . Sawin (2026) subsequently made the bound explicit and proved that, for arbitrarily large , more than unit-distance pairs occur. The construction uses algebraic number fields , infinite class field towers , and the Golod-Shafarevich theorem , and gives an asymptotic counterexample to square grid optimality.
Supports. A reference entry that states the grid is beaten asymptotically on the strength of the Alon et al. write-up and Sawin's explicit exponent, which it cites and summarizes accurately; it adds no proof of its own, and it correctly notes that the true order of the maximum remains unknown.
Not found in the text 7
The conjectured matching bound, in the form u(n)<=n^(1+o(1)), where o(1) denotes a quantity tending to 0 in little-O notation, or more concretely u(n)<=n^(1+C/lnlnn) for some absolute constant C, was refuted by an OpenAI-generated proof (Alon et al. 2026, OpenAI 2026).
Supports. A reference entry that restates the 2026 disproof accurately, crediting the OpenAI-generated proof, the Alon et al. write-up, and Sawin's explicit exponent; it adds no evidence of its own beyond finite illustrative pieces of the construction, and correctly notes that the true order of the maximum remains unknown.
Erdős (1946) proved a lower bound of u(n)=n^(1+Omega(1/lnlnn)), where Omega is big-Omega notation
Without evidence. 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.
Sawin (2026) subsequently made the bound explicit and proved that, for arbitrarily large n, more than n^(1.014) unit-distance pairs occur.
Without evidence. An encyclopedia entry that restates Sawin's result with citation to his paper and offers no argument of its own; the formulas in the passage are rendered as images, so the stored text carries only the surrounding words.
However, the true order of u(n) remains unknown. The best upper bound currently known is u(n)=O(n^(4/3)), where O is big-O notation (Szemerédi 2016).
Not yet read by this claim’s Steward.
The best upper bound currently known is u(n)=O(n^(4/3)), where O is big-O notation (Szemerédi 2016).
Without evidence. 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.
It is equivalent to finding a maximally dense unit-distance graph on n vertices.
Without evidence. The encyclopedia entry states the reformulation as a matter of definition and offers no argument for it; none is needed, since the two formulations differ only in vocabulary.
For every positive integer n, they constructed a set P of n points such that every subset A subset= P with at least two points has some distance occurring at least c|A|^2/n^(1-delta) times, for absolute constants c,delta>0.
Not yet read by this claim’s Steward.
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Record
- Kind
- unknown
- Authors
- Weisstein, Eric W.
- Publisher
- Wolfram Research, Inc.
- Published
- Not recorded
- DOI
- None
- Link
- https://mathworld.wolfram.com/ErdosUnitDistanceProblem.html
- Archived copy
- https://web.archive.org/web/20261025074332/https://mathworld.wolfram.com/ErdosUnitDistanceProblem.html
- Stored text
- 7,152 characters, retrieved Sep 14, 2026 · 67 sections, 24 passages
- Claims extracted
- Not recorded
- Versions
- One known
- Copies
- None recorded
- Corrections
- None recorded
- Facts checked
- Nov 9, 2026
History 9
- Nov 9, 2026DividedSegmenter · Divided into 91 parts by its own structure
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- Sep 16, 2026ReadingSteward of “Erdős's unit distance problem is equivalent to…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “The maximum number of unit distances among n pl…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “The square grid is not asymptotically optimal f…” · supports, read whole
- Sep 16, 2026ReadingSteward of “For arbitrarily large n there exist n-point pla…” · asserts without evidence, read whole
- Sep 16, 2026ReadingSteward of “The maximum number of unit distances among n po…” · supports, read whole
- Sep 16, 2026ReadingSteward of “The maximum number of unit distances among n pl…” · asserts without evidence, read whole