  We say a family of sets is intersecting if any two of its sets intersect, and
we say it is trivially intersecting if there is an element which appears in
every set of the family. In this paper we study the maximum size of a
non-trivially intersecting family in a natural "multi-part" setting. Here the
ground set is divided into parts, and one considers families of sets whose
intersection with each part is of a prescribed size. Our work is motivated by
classical results in the single-part setting due to Erd\H{o}s, Ko and Rado, and
Hilton and Milner, and by a theorem of Frankl concerning intersecting families
in this multi-part setting. In the case where the part sizes are sufficiently
large we determine the maximum size of a non-trivially intersecting multi-part
family, disproving a conjecture of Alon and Katona.
