Lovász theorem on Kneser graphs ($\chi (KG(n,k))=n-2k+2$)

For integers $n\ge 2k\ge 2$, the chromatic number of the Kneser graph is $n-2k+2$. The upper bound colours by the least element among the first $n-2k+1$ elements, with one final colour for all other sets. The Gale hemisphere lemma and the Lusternik-Schnirelmann-Borsuk theorem give the matching lower bound.