In this previous blof post I described the resolution by AI of a problem about primitive sets of integers. Now AI made a substancial step further and disproved famous unit distance conjecture of Paul Erdős. This was one of the central open problem in discrete geometry, and its resolution deserves publication in a jounral of the highest level, such as Annals of Mathematics.
For a finite set of points in the plane, let denote the number of unordered pairs such that . Let Equivalently, since we may rescale the whole configuration, is the maximum possible number of occurrences of the same distance among points in the plane.
In 1946, Erdős (Erdős 1946) asked how large can be. He observed that . This follows from the fact that the unit distance graph cannot contain a copy of , since two unit circles meet in at most two points. In 1984, Spencer, Szemerédi and Trotter (Spencer et al. 1984) improved this to , and this remains the state of the art for the upper bound.
On the other hand, Erdős showed that the square grid construction gives for some and infinitely many values of . To achieve this, one takes a grid with and chooses an integer with many representations as a sum of two squares. Many pairs of grid points then have distance , and after rescaling these become unit distances. Erdős conjectured that the grid construction is close to the truth, namely that
In 2026, a counterexample to this conjecture was found by an internal model at OpenAI. A human-verified and simplified version of the proof was then presented in (Alon et al. 2026).
The main idea of the construction is a striking generalization of Erdős’s original grid example. The ordinary square grid may be viewed as a box in the Gaussian integers , where distances are controlled by representations of integers as sums of two squares. The new construction replaces the Gaussian integers by rings of integers in suitable number fields . The crucial additional idea is that the fields are allowed to vary.
The proof in (Alon et al. 2026) made no attempts to optimize . Sawin (Sawin 2026) proved that one can take . This still leaves a large gap between the lower bound and the best known upper bound .