OpenAI Model Disproves Erdős's 1946 Unit Distance Conjecture

"The following result is due to an internal model at OpenAI." That is the disclosure on the first page of a short mathematics note posted to OpenAI's CDN about a conjecture Paul Erdős made in 1946.

The abstract calls the document "a short, digested, human-verified version of the recent OpenAI-generated counterexample to the Erdős unit distance conjecture." Erdős had asked how many pairs of n points in the plane can be exactly one unit apart, and conjectured at most n^{1+o(1)}. The model produced a construction beating that ceiling by an explicit polynomial factor.

The margin was small. Scott Aaronson wrote on his blog that the model's own construction gave n^{1+ε} unit-distance pairs "for ε ~ 10-38." A human, Will Sawin at Princeton, then pushed the same idea to roughly n^{1.014} in a companion paper. The best known upper bound is still n^{4/3}.

Aaronson also described how it came out. "My former student Lijie Chen simply gave GPT the problem, then GPT thought for a while and output a several-page argument that, on analysis by human experts, turned out to be correct," he wrote. The remarks paper is careful about what the humans did: it presents "a human-digested, somewhat simplified, and somewhat generalized version of the AI proof," attributing the underlying machinery in retrospect to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.

The note does not name which OpenAI model produced the argument. Five of the researchers we track had circulated the CDN link within days of it going up.