Matching and intersection problems for non-trivial $r$-partite $r$-uniform hypergraphs

A central theme in extremal combinatorics is the study of the maximum number of edges in an $r$-uniform hypergraph ($r$-graph) with matching number at most $s$ (the Erd\H{o}s Matching Conjecture) or with pairwise intersection at least $t$ (the $t$-intersection problem). The maximum sizes for these problems are typically achieved by trivial constructions: for the matching problem, the extremal construction consists of all edges intersecting a fixed set of $s$ vertices, while for the intersection problem, it consists of all edges containing a fixed set of $t$ vertices. In this paper, we investigate the \emph{non-trivial} $r$-partite $r$-graphs where each part is of size $n$. We determine the exact bounds for both the matching problem and the intersection problem when $n$ is sufficiently large. Furthermore, for the intersection problem, we resolve the cases $t=1$ and $t=r-2$ for all $n \ge 2$. Our results partially confirm a conjecture of Lu and Ma.