A new generalization of the Erdős-Ko-Rado theorem

Abstract

Let ℱ be a family ofk-subsets of ann-set. Lets be a fixed integer satisfyingk≦s≦3k. Suppose that forF ₁,F ₂,F ₃ ∈ ℱ |F ₁ ∪F ₂ ∪F ₃|≦s impliesF ₁ ∩F ₂ ∩F ₃ ≠ 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>n ₀(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.