Remarks on the disproof of the unit distance conjecture
Alon, Noga, Bloom, Thomas F., Gowers, W. T., Litt, Daniel, Sawin, Will, Shankar, Arul, Tsimerman, Jacob, Wang, Victor, Wood, Melanie MatchettarXiv.orgPublished May 20, 2026Original︎ ↗Archived copy
Stewards’ notes · 3
- The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Supports. The primary mathematical record of the disproof. Section 2 proves the theorem from two lemmas, a geometry-of-numbers count and a class-group pigeonhole, plus the existence of Golod–Shafarevich towers with a split prime; the authors describe the original AI proof as valid and their version as simplified and generalized. The evidence is the proof itself, which is short enough to check. 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. 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. Sep 16, 2026
- The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Supports. The primary mathematical record of the disproof. Section 2 gives a full proof from two lemmas (a geometry-of-numbers count and a class-group pigeonhole) plus the Golod-Shafarevich existence of suitable towers, with an explicit if tiny exponent; the authors describe the original AI proof as valid and their version as simplified and generalized. The evidence is the proof itself, which is short enough to check. 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.
Text54 passages · 0 marked
Not found in the text 4
There exists ε > 0 such that the following holds. There exists a sequence of point sets P_i in R^2 such that |P_i| → ∞ and the number of unit distances in P_i is at least |P_i|^{1+ε} for all i.
Supports. The primary mathematical record of the disproof. Section 2 gives a full proof from two lemmas (a geometry-of-numbers count and a class-group pigeonhole) plus the Golod-Shafarevich existence of suitable towers, with an explicit if tiny exponent; the authors describe the original AI proof as valid and their version as simplified and generalized. The evidence is the proof itself, which is short enough to check.
Erdős proved that $U(n) \geq n^{1+\Omega(1/\log\log n)}$ , and the best known upper bound is $U(n) \leq O(n^{4/3})$ .
Without evidence. 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.
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].
Not yet read by this claim’s Steward.
There exists ε > 0 such that the following holds. There exists a sequence of point sets P_i in R^2 such that |P_i| → ∞ and the number of unit distances in P_i is at least |P_i|^{1+ε} for all i.
Supports. The primary mathematical record of the disproof. Section 2 proves the theorem from two lemmas, a geometry-of-numbers count and a class-group pigeonhole, plus the existence of Golod–Shafarevich towers with a split prime; the authors describe the original AI proof as valid and their version as simplified and generalized. The evidence is the proof itself, which is short enough to check.
Lineage across 4 claims
Record
- Kind
- unknown
- Authors
- Alon, Noga, Bloom, Thomas F., Gowers, W. T., Litt, Daniel, Sawin, Will, Shankar, Arul, Tsimerman, Jacob, Wang, Victor, Wood, Melanie Matchett
- Publisher
- arXiv.org
- Published
- May 20, 2026
- DOI
- None
- Link
- https://arxiv.org/abs/2605.20695
- Archived copy
- https://web.archive.org/web/20261027190346/https://arxiv.org/abs/2605.20695
- Stored text
- 3,807 characters, retrieved Sep 16, 2026 · 44 sections, 54 passages
- Claims extracted
- Not recorded
- Versions
- One known
- Copies
- Copy of
- Corrections
- None recorded
- Facts checked
- Nov 9, 2026
History 7
- Nov 9, 2026DividedSegmenter · Divided into 98 parts by its own structure
- Nov 9, 2026FactsFacts fetcher · Facts looked up
- Sep 16, 2026ReadingSteward of “OpenAI's AI-generated disproof of the unit dist…” · supports, 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
- Sep 16, 2026ReadingSteward of “The maximum number of unit distances among n po…” · supports, read whole
- Sep 16, 2026Entered the graphEntered the graph; copy stored