What is the current mathematical level of AI?
Current AI models have seemingly impressive math capabilities—both Google and OpenAI won the IMO gold medal this year. But are they actually useful in math research? That’s still debated.
I want to contribute to this discussion with the following clear-cut experiment. I will take a clearly defined open problem in mathematics and give it to an AI model. The only input to the model is the problem itself, possibly with some generic prompting (“You are a research mathematician…”). Can AI already solve such open problems?
It can. I tried this for about 50 problems with GPT-5 and got a few results I consider nontrivial. I will share two examples in combinatorics; modulo the problem statement, the following solutions are 100% AI, including writeup.
It should be said that the following two questions can only barely be called “open problems”. The questions are not well-known and not even conjectured in a peer-reviewed article. I know one problem from a booklet of open problems and the other from personal communication. Yet these are questions that a concrete person working in the relevant area wondered about and did not immediately have a solution for. Of course, I am certain they would likely solve them, if they spent more time.
But look: beginnings always look like this. Take the first program that beat a chess grandmaster. The legitimate reaction to this event was that the grandmaster was “badly off form.” Yet, Kasparov has been beaten less than 10 years later. Similarly, I expect that in the coming months we will see examples that look more and more legitimate and get dominated in <10 years from now (and maybe much sooner).
Note: I don’t know of other people showcasing examples of this exact kind. There are benchmarks like FrontierMath and Humanity’s last exam. At this point in time, they remain unsaturated and thus very useful benchmarks.
But these are still not problems that one would write a paper about if they solved it (otherwise, the relevant researchers would write that paper and submit it to the benchmark). I think we will soon see results that are paper-worthy and this exploration is about how far from that we are now.
Of course, many mathematicians use AI in their work, but it is typically hard to disentangle what part of the work is done by human and what part by the AI.
Tightness of a recent paper on 2D tiling
There was a recent nice paper about tiling the plane $\mathbb{R}^2$ with polygonal tiles. One of the results in that paper is that every tiling of the plane with polygonal tiles is weakly periodic: it can be partitioned into finitely many pieces, each of them periodic.
Can this theorem be improved? Can we prove that every polygonal tiling is simply periodic? It turns out that the answer is no. AI can find a specific example of a polygonal tiling that is weakly periodic but not periodic.
The example is here.
A counterexample to a problem about functions
Here’s a cool fact about functions that has applications in graph theory.
Consider two functions $f$ and $g$ from a set $E$ into a set $F$ such that $f(x) \neq g(x)$ for every $x \in E$. Suppose that there exists a positive integer $n$ such that for any element $z \in F$, either $\lvert f^{-1}(z) \rvert \le n$ or $\lvert g^{-1}(z) \rvert \le n$.
Then $E$ can be partitioned into $2n + 1$ subsets $E_1, E_2, \dots, E_{2n+1}$ such that $f(E_i) \cap g(E_i) = \emptyset$ for each $1 \le i \le 2n + 1$.
Can we generalize this theorem to more functions? It is unclear how many sets would be needed in the partition if we generalize from $2$ functions to $k$ functions.
A particularly brave question was asked at KAMAK, the yearly retreat of combinatorics researchers from Charles University (first problem in this booklet). Maybe $2n+1$ sets is still enough?
This turns out to be too optimistic. Given this problem, AI came up with a straightforward construction showing that if you generalize the problem to $k$ functions, you need to partition into at least $kn$ subsets.
Here’s the short proof (fully AI-written).
