Proved
Erdős’s grid construction ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Remarks on the Disproof of the Unit Distance Conjecture · §1.1, History of the problem; §2 opening →
Observed Oct 21, 2026
Bound · version 1
Erdős’s grid lower bound
Statement
There is c > 0 such that for every sufficiently large n, some set of n distinct Euclidean planar points determines at least n^(1+c/log log n) unordered unit-distance pairs.
Assumptions
- Euclidean plane; unordered pairs at distance exactly one.
- The exponent gain c/log log n tends to zero, so this is consistent with the near-linear conjecture.
Scope
A lower bound for every large n from a scaled square grid; its exponent gain vanishes as n grows, unlike the 2026 fixed gain δ.
Evidence
Source reports
Connections
In this problem · publication v6 · Selected context
Erdős unit-distance problem →
Related result · Proved
The classical lower bound: superlinear, but with an exponent gain that vanishes.
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Remarks on the Disproof of the Unit Distance Conjecture →
- §1.1, History of the problem; §2 opening → · Original passage ↗
Recorded wording
Erdős showed with a √n × √n grid that n points can have n^(1+Ω(1/log log n)) unit distances, and conjectured an upper bound n^(1+o(1)). The best upper bound, O(n^(4/3)), is due to Spencer, Szemerédi and Trotter (1984).
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Noga Alon · Thomas F. Bloom · W. T. Gowers · Daniel Litt · Will Sawin · Arul Shankar · Jacob Tsimerman · Victor Wang · Melanie Matchett Wood · Preprint · Publication date May 20, 2026
Revision & attribution
Source revision: 2026-05-20. Observed Oct 13, 2026.
The mathematical starting point is attributed to OpenAI’s model; the nine named authors supply exposition, checking, simplification and context.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Erdős’s grid construction ↗
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026 · Source passage →
Record details
- Claim ID
- claim-unit-distances-erdos-grid
- Statement version
- 1
- SHA-256
- 2d0abc6c2ac9a0fbe958a193cb7cd11f53c0e78938acd60786f6061eb2b78371