Borsuk's Conjecture in Dimension 63

The diameter of a set of points is the largest distance between two of them. Karol Borsuk asked how many pieces you need, if you cut a set so that each piece has a smaller diameter than the whole set.

Small cases show the idea. A line segment needs 2 pieces, so you cut it in the middle. A disk needs 3 pieces, as in the figure, and 2 pieces are not enough. The three corners of an equilateral triangle also need 3 pieces, because any two corners are at the full diameter.

A disk with diameter 1, cut into three equal pieces. The longest distance inside one piece is about 0.87.

This pattern continues. In $n$ dimensions, $n + 1$ points can all be at the same distance from each other, so $n + 1$ pieces are sometimes necessary. Borsuk's conjecture says that $n + 1$ pieces are always enough, for every set of bounded size. It holds in 1, 2 and 3 dimensions.

In 1993, Jeff Kahn and Gil Kalai showed that it fails in very high dimensions. So the question became how low the failure goes. Later examples reached 65 dimensions, and then 64.

A point in 63 dimensions is a list of 63 numbers. Distance uses the same formula as on paper. You add the squares of the differences, then take the square root. So these are ordinary points, only with longer lists.

Max Grinsztajn reports a set of 321 points in 63 dimensions, found with GPT-5.5 Pro. It has one strong property. Any part of the set with a smaller diameter holds at most 5 points.

The rest is arithmetic. The conjecture allows 64 pieces. But 64 pieces of at most 5 points hold at most $64 \times 5 = 320$ points, which is one too few for 321. So at least 65 pieces are needed, and the conjecture fails in dimension 63.

The points come from a large symmetric graph called $G_{2} (4)$ . A graph is a set of dots, called vertices, joined by lines, called edges. This one has 416 vertices, each joined to 100 others. Andriy Bondarenko placed these vertices as points in 65 dimensions, and Thomas Jenrich and Andries Brouwer took a part of them that fits in 64. The new step picks 320 points that fit in 63 dimensions, then adds one more point. A public program checks the finite facts exactly.

The smallest dimension where the conjecture fails is still unknown. For a finite set of points, 63 is the lowest known. OpenAI separately reports an infinite set in 9 dimensions that cannot be cut into 10 pieces. If this holds, the conjecture already fails in dimension 9.