We show that there are sets of n points in the plane with n arbitrarily large that contain more than n^1.014 pairs of points separated by a distance exactly 1. This improves on very recent work of a team at OpenAI, who proved the same result with an inexplicit exponent greater than 1, disproving a conjecture of Erdős.
The method is number-theoretic, relying on constructing algebraic number fields of large degree and small discriminant with many primes of small norm via a Golod-Shafarevich criterion argument.
Attribution in the paper: the underlying proof was produced by OpenAI researcher Lijie Chen using an internal model, with Mark Sellke and Mehtaab Sawhney verifying correctness.