LONDON, July 22, 2026 — A new look at an old problem in mathematics, the cap set puzzle, has yielded small but significant advances thanks to an artificial intelligence tool called FunSearch. The work, published in Nature in December 2023, improved known lower bounds for sets of points in high-dimensional spaces with no three in a line — a problem that had stumped human reasoners for years.
Before FunSearch stepped in, the best known lower bounds for small dimensions were: 1,010 points in dimension 6, 3,478 in dimension 7, and 17,555 in dimension 8. The algorithm nudged those numbers up — to 1,012, 3,489, and 17,642, respectively.
The gains are slight: two points, eleven points, eighty-seven points. But the encouraging part, as researchers see it, is that these numbers came from a machine, not a human guess. The cap set problem essentially asks how large a set can be in an n-dimensional space without containing three collinear points.
It is a classic in combinatorics, with roots in both pure math and computer science. For decades, the best lower bounds for small n were generated by careful human reasoning and exhaustive searches.
FunSearch broke through that barrier. The upper bound, proven by mathematician Jordan Ellenberg and his colleague Dion Gijswijt back in 2016, sits at roughly 2.756 to the power n. That means the lower bound still has a long way to go.
But early findings suggest these new explicit constructions — not just numbers, but actual arrangements of points — matter for computational applications, such as testing algorithms and designing error-correcting codes.
How FunSearch Did It
The algorithm evaluated about 2.5 million candidate solutions, the Nature paper reported. The computational cost was modest by deep learning standards, using only a few GPU-days. That efficiency is what excites mathematicians: the method can be reapplied to other hard combinatorial problems without requiring enormous computing resources.
Still, experts advise caution. The gains are small, and the upper bound remains far off.
The encouraging part is that machine learning showed it can extend the frontier of human knowledge in a domain where intuition often falters. Patients should — well, in this case, it is researchers who should talk to their computer scientists and begin testing these methods on other open problems. For now, the cap set puzzle is far from solved.
But the path forward is clearer. While no cure for the problem is in sight, the approach offers a new tool for exploring these high-dimensional spaces — and that, for math, is genuine hope.
Researchers encourage continued testing and verification of such AI-assisted constructions. As always, anyone working on related problems should consult the literature and their computational collaborators before drawing firm conclusions.
