Uniform s-Cross-Intersecting Families

In this paper we study a question related to the classical Erdős–Ko–Rado theorem, which states that any family of k-element subsets of the set [n] = {1, . . . , n} in which any two sets intersect, has cardinality at most (︀ n−1 k−1 )︀ . We say that two non-empty families are A,B ⊂ (︀[n] k )︀ are s-cross-intersecting, if for any A ∈ A, B ∈ B we have |A ∩ B| ≥ s. In this paper we determine the maximum of |A|+ |B| for all n. This generalizes a result of Hilton and Milner, who determined the maximum of |A|+ |B| for nonempty 1-cross-intersecting families. MSc classification: 05D05