Erdos-Ko-Rado theorems for uniform set-partition systems

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 (k−t)! ( n−tc c )( n−(t+1)c c ) · · · (n−(k−1)c 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.