Extremal problems concerning Kneser graphs

Let d and S? be two intersecting families of k-subsets of an n-element set. It is proven that l~.JuS?l <(;:i)+(;::) holds for n>f(3+,/?)k, and equality holds only if there exist two points a, b such that {a, b} n F# 0 for all FE d u g, For n=2k+o(Jj;) an example showing that in this case max 1 d u B 1 = (1 o( 1 ))( ;) is given. This disproves an old conjecture of Erdiis [7]. In the second part we deal with several generalizations of Kneser’s conjecture.