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{\bf Karen Meagher and Lucia Moura }
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{\bf {E}rd\H{o}s-{K}o-{R}ado theorems for uniform set-partition systems}
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Two set partitions of an $n$-set are said to {\it $t$-intersect} if they
have $t$ classes in common. A {\it $k$-partition} is a set partition with
$k$ classes and a $k$-partition is said to be {\it uniform} if
every class has the same cardinality $c=n/k$.  In this paper, we prove
a higher order generalization of the {E}rd\H{o}s-{K}o-{R}ado 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.


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