Kneser's conjecture, chromatic number, and homotopy (Q755592)

The author proves a result on the simplicial complex formed by the neighborhoods of points of a graph and applies that to obtain an elegant proof of the Kneser conjecture of 1955 asserting that if the \(n\)-subsets of a \((2n+k)\)-element set is split into \(k+1\) classes, then one of the classes will contain two disjoint \(n\)-subsets. The proof depends on the theorem of \textit{K. Borsuk} [Fundam. Math. 20, 177--190 (1933; Zbl 0006.42403; JFM 59.0560.01)] that if the \(k\)-dimensional unit sphere is covered by \(k+1\) closed sets, then one of these contains two antipodal points.