Uniform s -Cross-Intersecting Families

Source abstract

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 $\binom{n-1}{k-1}$ . We say that two non-empty families ${\mathcal A}, {\mathcal B}\subset \binom{[n]}{k}$ are s-cross-intersecting if, for any A ∈ ${\mathcal A}$ , B ∈ ${\mathcal B}$ , we have | A ∩ B | ≥ s . In this paper we determine the maximum of | ${\mathcal A}$ |+| ${\mathcal B}$ | for all n . This generalizes a result of Hilton and Milner, who determined the maximum of | ${\mathcal A}$ |+| ${\mathcal B}$ | for non-empty 1-cross-intersecting families.