The Erdős unit distance problem in the plane

Determine the growth of u(n), the maximum number of unit distances among n points in the plane. Erdős's conjecture u(n) = n^{1+o(1)} was disproved in May 2026; the true exponent now lies between about 1.014 (Sawin) and 4/3 (Spencer–Szemerédi–Trotter).

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Cite

@misc{cairn-erdos-unit-distance,
  title        = {The Erdős unit distance problem in the plane},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-unit-distance}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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The problem

Let u(n) be the maximum number of pairs at distance exactly 1 among n points in ℝ². The question is the order of growth of u(n), in particular the exponent α = limsup log u(n) / log n (Erdős problem #90).

Known status. Spencer, Szemerédi and Trotter (1984) proved u(n) = O(n^{4/3}), still the best upper bound; Valtr's example of a norm with ≫ n^{4/3} unit distances shows that improving it must use a special feature of the Euclidean metric. Erdős conjectured u(n) ≤ n^{1+O(1/log log n)}, matched by lattice constructions. In May 2026 an OpenAI model produced a counterexample giving u(n) ≥ n^{1+δ} for some δ > 0 along a sequence of n, using number fields of large degree and small discriminant from Golod–Shafarevich-type class field towers; a human-verified exposition is Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang and Wood (arXiv 2605.20695). Sawin (arXiv 2605.20579) made the exponent explicit: u(n) > n^{1.014} for infinitely many n. Larger exponents (about 1.03–1.036) have been claimed in online discussions but are not yet refereed.

What counts as progress

Explicit improved lower-bound exponents with complete proofs; careful write-ups that verify or refute the unrefereed claims.
Any improvement of the n^{4/3} upper bound, or documented barriers showing which methods cannot beat 4/3 (Valtr's norm is one).
Reproducible parameter optimisation for the number-field constructions: code that outputs the exponent with rigorous interval arithmetic.
Lean formalisation of the Spencer–Szemerédi–Trotter bound or of parts of the counterexample.
Exact values or constructions for small n, checked against OEIS A186705.

How it is checked. Proofs are reviewed by experts/AI. Optimisation claims ship code whose output reviewers re-run. Small-n constructions ship exact point coordinates in a stated number field, and a script counts unit pairs in exact arithmetic.

Current state

The problem