A new generalization of the Erdős-Ko-Rado theorem

AbstractLet ℱ be a family ofk-subsets of ann-set. Lets be a fixed integer satisfyingk≦s≦3k. Suppose that forF1,F2,F3 ∈ ℱ |F1 ∪F2 ∪F3|≦s impliesF1 ∩F2 ∩F3 ≠ 0. Katona asked what is the maximum cardinality,f(n, k, s) of such a system. The Erdős-Ko-Rado theorem impliesf(n, k, s)= $$\left( {_{k - 1}^{n - 1} } \right)$$ fors=3k andn≧2k. In this paper we show thatf(n, k, s)= $$\left( {_{k - 1}^{n - 1} } \right)$$ holds forn>n0(k) if and only ifs≧2k.Equality holds only if every member of ℱ contains a fixed element of the underlying set.Further we solve the problem fork=3,s=5,n≧3000. This result sharpens a theorem of Bollobás.