https://palomar-registry.org/ New and updated Lean-verified results accepted by Palomar. en Mon, 03 Aug 2026 08:37:32 +0000 https://palomar-registry.org/entry.html?id=PALOMAR-2026-07-29-000001&version=3 https://palomar-registry.org/entry.html?id=PALOMAR-2026-07-29-000001&version=3 For every C > 0 and every N there exist n ≥ N and an n-point set P ⊆ ℝ² whose number of unit-distance pairs exceeds n^(1 + C / log log n). This refutes Erdős's 1946 conjectured bound ν(n) ≤ n^(1 + C/log log n) for an absolute constant C. The theorem quantifies over C > 0; this is harmless because any bound with C ≤ 0 would imply the corresponding bound for a positive C. Sat, 01 Aug 2026 12:57:32 +0000 math.CO 52C10