BREAKING NEWS: OpenAI has disproved Erdős' unit-distance conjecture

Added: 2026-05-23

[00:00:00]Huge breaking news in the world of AI and mathematics because OpenAI just announced that they have disproved a famous conjecture in combinatorics, in discrete geometry. The problem itself is a conjecture of Erdős, and in fact, this was one of his favorite problems to work on. The problem is Erdős' unit distance problem that asks how often can the same distance occur among n points in the plane, and experts in combinatorics like Noga Alon have described this as one of the best-known open problems in combinatorics, and that means that a lot of people are aware and have worked on this problem, including Erdős. You can fix the distance in the problem to be one, and then it becomes the unit distance problem. How can you arrange n points in the plane so that there are as many points as possible that are one unit apart from each other. Erdős thought that this kind of configuration was the best possible, and it gave you some sort of upper bound on how many points can be a unit apart. However, the discovery by OpenAI is that Erdős' bound is not true, and there are, in fact, configurations similar to this one that beat that bound. By the way, I also made this graph with ChatGPT. So, what do the experts say about the proof that the LLM has come up with? For example, Noga Alon, who is an expert in combinatorics, says that this proof is an outstanding achievement. Jacob Zimmerman, who might get a Fields Medal this year, says that this is a really impressive piece of work, and I would accept it for any journal without hesitation. And he continues that the AIs or LLMs have an edge because they can try all known methods, and they can also play with it longer than mathematicians without getting overwhelmed. Daniel Litt makes some very interesting points here because the problem is in combinatorics, but the proof itself uses some sophisticated algebraic number theory.

[00:02:04]So for a human to discover this proof, they would have had to be experts both in combinatorics and in algebraic number theory because people in algebraic number theory like he or I had never heard of this problem. Now other experts like Thomas Bloom, who is the person who has curated the collection of Erdos' problems online, is not so enthusiastic about the proof itself because he says that it is highly non-trivial, but at the end it's just a generalization of the original lattice based construction of Erdos. Melanie Matchett Wood, who is a Harvard professor and also a MacArthur fellow, says that it is easy to jump to hasty conclusions about what this means because if we had assembled a number of mathematicians to work for a month on this particular problem and these mathematicians had a broad set of skills and backgrounds, then they would have found the same counterexample.

[00:02:57]Similarly, Arora Shankar in his comments says that this is a very human proof.

[00:03:02]This is not an alien proof, some proof that includes some new idea that nobody has ever seen. The methods that are being used here are methods that are well-known.

[00:03:15]So yes, it is a very ingenious proof, but it is not a paradigm change in what AI can do. Finally, I'll say that this is very impressive, but there is a company that is throwing a lot of money to solve these problems and this is one that they were able to solve, but I can assure you that there are many other problems that have some name attached to it that are much more famous than this problem that these companies are trying to solve and they are not being able to solve them.

[00:03:48]Whether we are going to get there and OpenAI or other company will eventually solve a millennial problem, we don't know. But for now, they are just telling you, heralding all their victories, but they're not telling us all the failures.