  Let $\mathcal{A}\subseteq{[n]\choose a}$ and $\mathcal{B}\subseteq{[n]\choose
b}$ be two families of subsets of $[n]$, we say $\mathcal{A}$ and $\mathcal{B}$
are cross-intersecting if $A\cap B\neq \emptyset$ for all $A\in\mathcal{A}$,
$B\in\mathcal{B}$. In this paper, we study cross-intersecting families in the
multi-part setting. By characterizing the independent sets of vertex-transitive
graphs and their direct products, we determine the sizes and structures of
maximum-sized multi-part cross-intersecting families. This generalizes the
results of Hilton's and Frankl--Tohushige's on cross-intersecting families in
the single-part setting.
