The 0/1-borsuk Conjecture Is Generically True for Each Fixed Diameter

In 1933 Karol Borsuk asked whether every compact subset of R can be decomposed into d + 1 subsets of strictly smaller diameter. The 0/1Borsuk conjecture asks a similar question using subsets of the vertices of a ddimensional cube. Although counterexamples to both conjectures are known, we show in this article that the 0/1-Borsuk conjecture is true when d is much larger than the diameter of the subset of vertices. In particular, for every k, there is a constant n which depends only on k such that for all configurations of dimension d > n and diameter 2k, the set can be partitioned into d− 2k+2 subsets of strictly smaller diameter. Finally, Lásló Lovász’s theorem about the chromatic number of Kneser’s graphs shows that this bound is in fact sharp.