Partial covering arrays and a generalized Erdos-Ko-Rado property

The classical Erd\H os-Ko-Rado theorem states that if $k\le \text{floor}n\mathrm{/}2$ then the largest family of pairwise intersecting $k$-subsets of $[n]=\{0,1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ is of size $\left(\genfrac{}{}{0px}{}{n-1}{k-1}\right)$. 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 $\mathcal{A}$ of $k$-subsets of $[n]$ that satisfies the following property: For each $A,B,C\in \mathcal{A}$, each of the four sets $A\cap B\cap C;A\cap B\cap C^C; A\cap B^C\cap C; A^C\cap B\cap C$ 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.