IN FOUR DIMENSIONS, SEVEN PIECES ALWAYS DO

Take a shape — any bounded set of points — and call its diameter the largest distance between two of its points. Now cut it into pieces so that every piece is strictly narrower than the whole. What is the smallest number of pieces that always works, whatever the shape?

That number, in n dimensions, is called the Borsuk number b_n. In 1933, Borsuk asked whether it always equals n + 1. It cannot be smaller: the corners of a regular simplex — an equilateral triangle in the plane, a regular tetrahedron in space — need n + 1 pieces, since any two corners are a full diameter apart. And n + 1 is enough in one, two and three dimensions. In 1993, Kahn and Kalai showed the guess fails in sufficiently high dimensions.

The first open case

Four dimensions is the first place where nobody knows the answer. An old general bound by Lassak, 2ⁿ⁻¹ + 1, gives at most nine pieces there. Earlier in 2026, Tolmachev and Voronov brought it down to eight, with a computer-assisted construction.

A. Arman and A. Prymak of the University of Manitoba, A. Bondarenko of the Norwegian University of Science and Technology and D. Radchenko of the Institut des Hautes Études Scientifiques now prove a stronger statement. Every set of diameter 1 in four dimensions can be covered by seven balls, all with the same diameter, strictly less than 0.9999. Each ball is narrower than the set, so assigning each point to one ball cuts the set into seven narrower pieces: b₄ ≤ 7. The answer now lies somewhere between five and seven.

Start from the smallest enclosing ball

The proof begins with the circumball: the smallest ball containing the set. A classical theorem by Jung limits how big it can be — its radius is at most about 0.63 (its square at most 2/5) for a set of diameter 1. The set touches the circumball’s surface, and among those touching points at most five are enough to pin the ball in place; the authors call them contact points. They constrain where the rest of the set can be, since nothing lies more than 1 away from any of them.

The authors then split the problem into three cases, by the size of the circumball:

Small circumball: seven balls in a fixed, explicit arrangement cover the entire circumball, whatever the set inside. This is proved by hand.
Large circumball: there are five contact points. Five balls, each centred at 9/20 of the way from the centre to one contact point, cover almost everything; two more balls cover what remains, two cone-shaped regions. Also proved by hand.
In between: three contact points forming an acute triangle reduce every possible set to a family of shapes described by just four numbers. That is where the computer comes in.

23,494 boxes, no rounding

The four numbers fill a four-dimensional cube of possibilities. For each small box of that cube, a numerical search proposes seven centres. Then an exact test checks that the same seven balls cover every shape in the box.

The trick is a classic of computational geometry: adding a fifth coordinate equal to the squared distance from the origin turns “inside this ball” and “outside that ball” into flat (linear) inequalities. The question becomes whether a polytope — a shape with flat faces — cut out by twelve such inequalities sits entirely above a curved bowl — and for that, it is enough to check its corners. The program examines all 792 candidate corners, using only whole numbers and exact fractions, with no numerical tolerance.

When a box fails, it is cut in two and both halves are tried again. In the end, the cube was cut into 23,494 boxes, down to 29 levels of halving, and every one passed. Their volumes add up to exactly 1: nothing is left out. The data and verification program are public. The authors note that the program uses exact arithmetic but is not a formal proof checked by a proof assistant.

How the idea arrived

The authors are open about how they got there. An earlier version of their work only reached eight, and needed far more computation; language models then helped only with code. They later explored the problem with ChatGPT Astra, first looking for purely analytic arguments. That exploration led to the approach built on the circumball, which shrank the computer search considerably. They also used ChatGPT for coding and editing, and checked every mathematical argument and the verification code themselves.

Borsuk’s own guess for four dimensions, five pieces, remains neither proved nor refuted. Two steps separate the new bound from it.