Karol Borsuk is most famous for the Borsuk-Ulam theorem, discussed in his ground-breaking paper *Drei Sätze über die *n*-dimensionale euklidische Sphäre*. The same paper also mentions the following conjecture:

Can every bounded subset of the space R^{n} be partitioned into (*n*+1) sets, each of which has a smaller diameter?

Informally, can you cut a *n*-dimensional shape into (*n*+1) pieces which are “smaller”, in the sense that the diameter (the maximum distance between two points) decreases? This is an example of a planar shape which fits the conjecture:

Borsuk’s conjecture seems very obvious. Yet, progress was made only slowly: Borsuk himself proved the case *n*=2 in 1931, Perkal the case *n*=3 in 1947, Hadwiger for all *n* but only for smooth convex bodies, Riesling for all centrally symmetric bodies and Dekster for all bodies of revolution. No final proof, however, was found.

The problem was finally solved in 1993 by Jeff Kahn and Gil Kalai. They demonstrated the conjecture of Borsuk actually is *false*, by constructing a 1325-dimensional counterexample!

The best bound currently known is due to Thomas Jenrich, with *n*=64*.*