Extremal problems concerning Kneser graphs

Abstract

Let A and B be two intersecting families of k-subsets of an n-element set. It is proven that | A ∪ B | ≤ ( k−1 n−1 ) + ( k−1 n−1 ) holds for n> 1 2 (3+ 5 )k , and equality holds only if there exist two points a, b such that { a, b} ∩ F ≠ ∅ for all F ∈ A ∪ B . For n = 2k + o( k ) an example showing that in this case max | A ∪ B | = (1−o(1))( k n ) is given. This disproves an old conjecture of Erdös [7]. In the second part we deal with several generalizations of Kneser's conjecture.