Uniform s-Cross-Intersecting Families

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.