A combinatorial conjecture formulated by Kneser (1955). It states that whenever the -subsets of a -set are divided into 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 -subsets, and where each edge connects two disjoint subsets, is not -colorable. More precisely, his results says that the chromatic number is equal to , and this implies that Kneser's conjecture is always false if the number of classes is increased to .
An alternate proof was given by Bárány (1978).