We are given an $n$-partite graph $G$. Each partition has $n$ vertices, some of which may be isolated. Let us number the vertices in some $i^{th}$ partition as $V_{i1},V_{i2},...,V_{in}$.

Now each non-isolated vertex $V_{ij}$ has at least one neighbor in each of the remaining $n-1$ partitions s.t. for a given numbering of the vertices, the $n$ vertices (vertex $V_{ij}$ and its $n-1$ neighbors) form a permutation on the second index. For e.g., consider a 4-partite graph. Each partition has 4 vertices.

Using the numbering as given above, vertex $V_{12}$ has as neighbors vertices $V_{21},V_{34},V_{43}$. Vertex $V_{12}$ can have other neighbors as well. We need to show that the graph $G$ will always contain $n$-clique.

A stronger claim would be to say that every vertex is part of some $n$-clique.