Amazing: Erdős’ Unit Distance Problem was Disproved! It was achieved by AI!

Paul Erdős’s, in his 1946  paper published in the American Mathematical Monthly, posed two general questions about the distribution of distances determined by a finite set of points in a metric space.

1. Unit Distance Problem: At most how many times can the same distance (say, distance 1) occur among a set of n points?

2. Distinct Distances Problem: What is the minimum number of distinct distances determined by a set of n points?

Erdős conjectured that in the plane the number of unit distances determined by n points is at most , for a positive constant c, but the best known upper bound, due to Spencer, Szemeredi, and Trotter is only .

As for the Distinct Distances Problem, the order of magnitude of the conjectured minimum is .  In 2010 a sensational paper of Guth and Katz presents a proof of an almost tight lower bound of the order of Janos Pach wrote about it in this 2010 post. See also this post from 2008, that explains (among other things) the proof for the upper bound.

I have just learned that an internal model of OpenAI have very recently disproved Erdős’ conjecture for the unit distance problem and found an example with more than unit distances. This is truly amazing. The construction relies on algebraic number theory.

Here is Open AI announcement and technical paper.

The proof is presented, explained and discussed in the paper Remarks on the disproof of the unit distance conjecture by Alon, Bloom, Gowers, Litt, Sawin, Shankar,Tsimerman, Wang, and Matchett Wood. (I learned about it from an answer to an MO question on the topic of mathematics and AI.)

Like the computer-based proof of the four color theorem in 1976 by Appel and Haken, this may well be a scientific landmark whose importance goes beyond combinatorics and beyond mathematics. I will add a link to the Open AI document when I will have it. (Done.)

There is also a short Open AI video where the result is presented by Tim Gowers and by four members of the team, Lijie Chen, Mark Sellke, Mehtaab Sawhney, and Sebastien (Seb) Bubeck.

Updates: Will Sawin proved a lower bound of This was further improved to ; There are interesting comments on the ErdosProblems site; The list of best constants so far can be found in Terence Tao’s Github. Sawin also proved in the same paper a limit 1.2143 for the new method.

The result is reported and discussed on the Geomblog, computational complexity, and Shtetl Optimized; There is an interesting commentary by Erik Hoel. Following the OpenAI result, Anthropic reported being able to autonomously disprove the unit-distance conjecture (in its strongest form) with its system. Inspired by the disproof of Erdős’ unit distance conjecture, Thomas Bloom, Will Sawin, Carl Schildkraut, and Dmitrii Zhelezov disproved Erdős-Szemeredi Sum-Product Conjecture (over the reals). (See the 2008 post mentioned above for the statement and connection.) Cosmin Pohoata whote a detailed beautiful post on the sum-product breakthrough. Thomas Bloom wrote a detailed beautiful blog post about the developments.

Remarks (about the problem): For an approach to improve the upper bound for the problem see the paper Erdős’s unit distance problem and rigidity, by János Pach, Orit E. Raz, József Solymosi  (2025). We discussed a lecture by Orit Raz on the topic in this post. Other posts (in “Combinatorics and more”) related to the Erdos distance problem and to related problems are listed in this comment.

Remarks (About the AI+math): This breakthrough adds to several others remarkable applications of AI to Math. See for example this post over Terry Tao’s blog on Erdős problem 1196 which was settled by Open AI model a few weeks ago. Very recently, a team at DeepMind  used their model to solve several open problems and their proofs come with Lean verification.  From ancient times (January 2026) let me mention an interesting blog post Problem 728 and the use of AI on Erdős problems, by Kevin Baretto. (June 3, 2026) There is an interesting Leiden Declaration on Artificial Intelligence and Mathematics.

Closer to me personally: Ori Gurel-Gurevich, Asaf Nachmias, and Sushant Sachdeva used an internal model of OpenAI to settle a problem about electrical flow in graphs; (I just met Asaf a few days ago in a special Day of Tel-Aviv University devoted to AI and Math.) Pablo Soberón used  LLM-based tools in the brainstorming stage of a recent truly remarkable paper: Tverberg cores and Kalai’s cascade conjecture.