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Kneser’s conjecture, chromatic number, and homotopy. (English) Zbl 0418.05028
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 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.

MSC:
05C15 Coloring of graphs and hypergraphs
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References:
[1] Borsuk, K, Drei Sätze über die n-dimensionale euklidische sphäre, Fund. math., 20, 177-190, (1933) · JFM 59.0560.01
[2] Erdös, P; Hajnal, A, On chromatic number of graphs and set-systems, Acta math. acad. sci. hungar., 17, 61-99, (1966) · Zbl 0151.33701
[3] Erdös, P; Hajnal, A, Kromatikus gráfokról, Mat. lapok, 18, 1-4, (1967)
[4] Garey, M; Johnson, D.S, The complexity of near-optimal graph coloring, J. assoc. comput. Mach., 23, 43-49, (1976) · Zbl 0322.05111
[5] Stahl, S, n-tuple colorings and associated graphs, J. combinatorial theory ser. B, 20, 185-203, (1976) · Zbl 0293.05115
[6] Kneser, M, Aufgabe 300, Jber. Deutsch. math.-verein., 58, (1955)
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