Unit Distance Problem
An unnamed OpenAI reasoning model disproved Erdős's 1946 conjecture, building planar point sets with n^(1+δ) unit-distance pairs.
Let u(n) be the maximum number of pairs at distance exactly 1 among n points in the plane — Erdős Problem #90, first asked in 1946. The square grid achieves n^(1+c/log log n) such pairs, and Erdős conjectured this near-linear count was essentially optimal: u(n) = n^(1+o(1)). The famous upper bound u(n) = O(n^(4/3)) of Spencer, Szemerédi and Trotter (1984) is still the standing ceiling. In May 2026 an internal, unnamed OpenAI general-purpose reasoning model disproved the conjecture by a genuinely new construction: it built point sets from totally real number fields of growing degree (Golod–Shafarevich class field towers in which fixed primes split completely), yielding u(n) ≥ n^(1+δ) for a fixed δ > 0 and infinitely many n — a polynomial improvement over the grid. Nine mathematicians (Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, Matchett Wood) verified the argument and posted a companion paper; Will Sawin then made the exponent explicit at u(n) > n^(1.014), later pushed past n^(1.03). The disproof settles the direction of the conjecture but does not close the wide gap up to the open O(n^(4/3)) bound.
Formula
Maximum number of point pairs at exactly distance 1 among n points in the plane.
Erdős believed the square grid was near-optimal — now known to be false.
Number-field point sets of growing degree beat the grid by a fixed polynomial factor.
The gap between n^(1+δ) and n^(4/3) remains the famous open unit-distance problem.
Summary
Let u(n) be the maximum number of pairs at distance exactly 1 among n points in the plane — Erdős Problem #90, first asked in 1946. The square grid achieves n^(1+c/log log n) such pairs, and Erdős conjectured this near-linear count was essentially optimal: u(n) = n^(1+o(1)). The famous upper bound u(n) = O(n^(4/3)) of Spencer, Szemerédi and Trotter (1984) is still the standing ceiling. In May 2026 an internal, unnamed OpenAI general-purpose reasoning model disproved the conjecture by a genuinely new construction: it built point sets from totally real number fields of growing degree (Golod–Shafarevich class field towers in which fixed primes split completely), yielding u(n) ≥ n^(1+δ) for a fixed δ > 0 and infinitely many n — a polynomial improvement over the grid. Nine mathematicians (Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, Wang, Matchett Wood) verified the argument and posted a companion paper; Will Sawin then made the exponent explicit at u(n) > n^(1.014), later pushed past n^(1.03). The disproof settles the direction of the conjecture but does not close the wide gap up to the open O(n^(4/3)) bound.
