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Two set partitions of an $n$-set are said to $t$-intersect if they have $t$ classes in common. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of the Erdős-Ko-Rado theorem for systems of pairwise $t$-intersecting uniform $k$-partitions of an $n$-set. We prove that for $n$ large enough, any such system contains at most $${1\over(k-t)!} {n-tc \choose c} {n-(t+1)c \choose c} \cdots {n-(k-1)c \choose c}$$ partitions and this bound is only attained by a trivially $t$-intersecting system. We also prove that for $t=1$, the result is valid for all $n$. We conclude with some conjectures on this and other types of intersecting partition systems.

Covering arrays with mixed alphabet sizes, or simply mixed covering arrays, are natural generalizations of covering arrays that are motivated by applications in software and network testing. A (mixed) covering array A of type Q k i¼1 g i is a k  N array with the cells of row i filled with elements from Z gi and having the property that for every two rows i and j and every ordered pair of elements ðe; f Þ 2 Z gi  Z gj , there exists at least one column c, 1 c N, such that A i;c ¼ e and A j;c ¼ f . The (mixed) covering array number, denoted by cað Q k i¼1 g i Þ, is the minimum N for which a covering array of type Q k i¼1 g i with N columns exists. In this paper, several constructions for mixed covering arrays are presented, and the mixed covering array numbers are determined for nearly all cases with k ¼ 4 and for a number of cases with k ¼ 5.

A Sperner partition system is a system of set partitions such that any two set partitions $P$ and $Q$ in the system have the property that for all classes $A$ of $P$ and all classes $B$ of $Q$, $A \not\subseteq B$ and $B \not\subseteq A$. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of Sperner's Theorem. In particular, we show that if $k$ divides $n$ the largest Sperner $k$-partition system on an $n$-set has cardinality ${n-1 \choose n/k-1}$ and is a uniform partition system. We give a bound on the cardinality of a Sperner $k$-partition system of an $n$-set for any $k$ and $n$.

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