Borsuk's conjecture

Borsuk's conjecture asserts that every bounded set in n-dimensional Euclidean space can be partitioned into n + 1 subsets, each with smaller diameter than the original set.

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Diameter: The diameter of a set is the greatest distance between any two of its points. The conjecture compares each part’s diameter directly with that of the original set.
Karol Borsuk: Karol Borsuk was a Polish mathematician whose work included topology and geometric partition problems. He posed the diameter-partition conjecture in 1933.
Borsuk's theorem in the plane: Every bounded subset of the Euclidean plane can be partitioned into three subsets of smaller diameter. The conjectured bound is true in dimension two.
Jeff Kahn: Jeff Kahn is an American mathematician known for work in combinatorics and probability. He co-authored the high-dimensional counterexample that refuted the conjecture.
Borsuk's conjecture counterexamples: Borsuk's conjecture counterexamples are bounded sets requiring more than dimension plus one smaller-diameter parts. Their existence overturned the conjecture’s universal claim.
Partition of a set: A partition divides a set into nonempty, pairwise disjoint subsets whose union is the set. The proposed bound concerns the number of pieces in such a division.
Borsuk's problem: Borsuk's problem asks how many smaller-diameter subsets are needed to partition a bounded set. It is the problem from which the conjecture takes its name.
Borsuk's theorem in three dimensions: Every bounded subset of three-dimensional Euclidean space can be partitioned into four subsets of smaller diameter. The conjectured bound is also true in dimension three.
Gil Kalai: Gil Kalai is an Israeli mathematician known for work in combinatorics and convex geometry. He co-authored the counterexample showing the conjecture fails in high dimensions.
Borsuk number asymptotics: Borsuk number asymptotics study how the required number of smaller-diameter parts grows with dimension. After the conjecture failed, bounding this growth became a central direction.