partition into smaller pieces · false above dim 63
↑ geometry pack · erdős · guthkatz · hadwiger · runner · kakeya · capset · szemerédi–trotter · heilbronn · viazovska
Borsuk's 1933 conjecture: every bounded set in ℝᵈ can be cut into d+1 pieces, each of strictly smaller diameter than the original. In dim 2 you need 3 pieces; in dim 3 you need 4. Drag the polygon to test — the radial partition from the centroid keeps every piece under the original diameter, no matter what you try.
Borsuk's conjecture stood for 60 years before Kahn and Kalai broke it in dim 1325 by reusing the Frankl–Wilson theorem. The minimum known dimension where the conjecture fails has been shrinking ever since — currently down to 64 (Jenrich 2014). Whether it's false in any dimension between 4 and 63 is still open.
Take a bounded subset of ℝᵈ. Its diameter is the largest distance between any two of its points. Borsuk asked: can we always split it into pieces, each of smaller diameter, and how many pieces do we need?
Borsuk 1933: d+1 pieces always suffice.
This isn't crazy. A disk in the plane splits into 3 sectors of 120° each; each sector has diameter √3·R < 2R. A ball in 3-space splits into 4 pieces using an inscribed regular tetrahedron. Borsuk himself proved it in dim 2; Eggleston proved it in dim 3 using a clever projection argument. The conjecture was on its way.
Sixty years later, Jeff Kahn and Gil Kalai exhibited a finite set of ±1 vectors in dim 1325 that resists partition into 1326 smaller-diameter pieces. The construction uses the Frankl–Wilson theorem on the rank of certain set-incidence matrices over finite fields. The "set" is the vertices of a Hamming-style configuration where most pairs of vectors achieve the maximum pairwise distance — so any partition must split many such pairs apart, and 1326 isn't enough.
The number of pieces required actually grows faster than any polynomial in d. Kahn and Kalai showed B(d) ≥ (1.2)√d asymptotically; later authors have improved the base.
Once the conjecture was known to fail somewhere, attention turned to the lowest dimension where it does. Each improvement comes from sharpening the combinatorial construction:
Whether the conjecture fails in any dimension between 4 and 63 is open. For applications in convex geometry it's also open whether smooth convex bodies satisfy the conjecture — the known counterexamples are all finite point sets, which is allowed but feels like cheating.