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

Patricia A. Carey·Anant P. Godbole
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Abstract

The classical Erd\H os-Ko-Rado theorem states that if $k\le\floor{n/2}$ then the largest family of pairwise intersecting $k$-subsets of $[n]=\{0,1,...,n\}$ is of size ${{n-1}\choose{k-1}}$. 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 ${\cal A}$ of $k$-subsets of $[n]$ that satisfies the following property: For each $A,B,C\in{\cal A}$, 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.

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Patricia A. Carey, Anant P. Godbole. 2005-12-06. Partial covering arrays and a generalized Erdos-Ko-Rado property. https://arxiv.org/abs/math/0512139

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