The companion paper provides a human-digested argument and background for the exponent gain, including the classical upper bound.
Source publication and attributed assessments are distinct from independent OpenMath review.
The exact passages cited in OpenMath. Each passage identifies how its wording was recorded.
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).
Passage ID: erdos-unit-distances:entry:remarks-history:v1
Source item: history
Source record: erdos-unit-distances:source:remarks:v1
The companion paper provides a human-digested argument and background for the exponent gain, including the classical upper bound.
Passage ID: erdos-unit-distances:entry:remarks:v1
Source item: remarks
Source record: erdos-unit-distances:source:remarks:v1
Explicit citations in versioned problem publications. A citation records a use; it does not confer proof or independent verification.
Problem publication v6
Source assessment
Proved
Erdős’s grid construction
Only the exact statement and passage cited here; not an independent OpenMath assessment.
Observed Oct 21, 2026
Statement
Source assessment
Proved
Classical 4/3 upper bound
Historical universal upper bound; not a claim of optimality.
Observed Oct 13, 2026
Source assessment
Disproved
Near-linear Euclidean conjecture refuted
The n^(1+o(1)) upper bound, not the exact extremal growth.
Source date May 20, 2026 · Observed Oct 13, 2026
Question
Erdős unit-distance problem
History
A model-generated construction refutes near-linear growth
History
Erdős introduces the planar distance question
Further reading
Read the setting and notation before comparing the results.
Recorded authorship
Noga Alon · Thomas F. Bloom · W. T. Gowers · Daniel Litt · Will Sawin · Arul Shankar · Jacob Tsimerman · Victor Wang · Melanie Matchett Wood
Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, Melanie Matchett Wood
The mathematical starting point is attributed to OpenAI’s model; the nine named authors supply exposition, checking, simplification and context.
OpenMath source records preserve exact citations. Observation dates order these records; they do not establish the latest upstream paper or proof.
Remarks on the Disproof of the Unit Distance Conjecture
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