Conjecture · version 1
For every there is N such that every finite set P of n ≥ N distinct Euclidean planar points determines at most unordered unit-distance pairs.
The near-linear asymptotic assertion is refuted; this does not determine the exact maximum.
Disproved
The n^(1+o(1)) upper bound, not the exact extremal growth.
Remarks on the Disproof of the Unit Distance Conjecture · Theorem 1.1; Sections 1.1–1.3 and 2 →
Source dated May 20, 2026 · Observed Oct 13, 2026
In this problem · publication v6 · Selected context
Primary formulation · Disproved
Relationship not independently reviewed. This concerns its use in this problem.
The maintained problem entry records the near-linear upper-bound question, its 1946 origin and the 2026 disproof, while retaining the unresolved extremal growth question.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
Erdős Problems · Project documentation
Source revision: Observed 2026-10-13. Observed Oct 13, 2026.
No specific AI contribution is attributed by this source record.
Theorem 1.1 constructs arbitrarily large planar sets with a fixed positive exponent gain in their unordered unit-distance count.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.
OpenAI · Preprint · Publication date May 20, 2026
Source revision: 2026-05-20. Observed Oct 13, 2026.
The manuscript credits an internal OpenAI model with the original argument; subsequent human editing and external mathematical checking are separate roles.
The companion paper provides a human-digested argument and background for the exponent gain, including the classical upper bound.
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
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.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Disproved
The n^(1+o(1)) upper bound, not the exact extremal growth.
Source dated May 20, 2026 · Observed Oct 13, 2026 · Source passage →