Partial Covering Arrays and a Generalized Erdős-Ko-Rado Property

The classical Erdős-Ko-Rado theorem states that if k ≤ ⌊n/2⌋ then the largest family of pairwise intersecting k-subsets of [n] = {0, 1, . . . , n} is of size ( n−1 k−1 ) . A family of k subsets satisfying this pairwise intersecting property is called an EKR family. We generalize the EKR property and provide asymptotic lower bounds on the size of the largest family A of k-subsets of [n] that satisfies the following property: For each A,B,C ∈ A, each of the four sets A ∩ B ∩ C;A ∩ B ∩ CC ;A ∩ BC ∩ C;AC ∩ B ∩ C are non-empty. This generalized EKR (GEKR) property is motivated, generalizations are suggested, and a comparison is made with fixed weight 3-covering arrays. Our techniques are probabilistic, and reminiscent of those used in [5] and in the work of Roux, as cited in [8].