Color the vertices of a triangulated triangle with three colors such that:
then there exists a small triangle whose vertices are colored with all three different colors. More precisely, there exists an odd number of such triangles.
This result looks playful and innocent but is in fact quite powerful. It is known, for instance, to lead to an easy proof of Brouwer’s fixed point theorem. Its power mainly lies in building bridges between discrete, combinatorial mathematics and continuous mathematics.
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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 Rn 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.