Kneser's Conjecture

A combinatorial conjecture formulated by Kneser (1955). It states that whenever the n-subsets of a (2n+k)-set are divided into k+1 classes, then two disjoint subsets end up in the same class.

Lovász (1978) gave a proof based on graph theory. In particular, he showed that the Kneser graph, whose vertices represent the n-subsets, and where each edge connects two disjoint subsets, is not (k+1)-colorable. More precisely, his results says that the chromatic number is equal to k+2, and this implies that Kneser's conjecture is always false if the number of classes is increased to k+2.

An alternate proof was given by Bárány (1978).

See also

Kneser Graph

This entry contributed by Margherita Barile

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Bárány, I. "A Short Proof of Kneser's Conjecture." J. Comb. Th. A 25, 325-326, 1978.Godsil, C. and Royle, G. Algebraic Graph Theory. New York: Springer-Verlag, p. 160, 2001.Kneser, M. "Aufgabe 300." Jahresber. Deutsch. Math.-Verein 58, 1955.Lovász, L. "Kneser's Conjecture, Chromatic Numbers and Homotopy." J. Comb. Th. A 25, 319-324, 1978.

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Kneser's Conjecture

Cite this as:

Barile, Margherita. "Kneser's Conjecture." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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