$r$-uniform hypergraphs and complete balanced $r$-partite $r$-graphs $K_{s,\dots ,s}^{(r)}$

For $r\ge 1$, an $r$-uniform hypergraph is a pair $(V,E)$ with $V$ finite and $E\subseteq [V{]}^r$. Its edges are $r$-element vertex sets. Ordinary subhypergraph containment means injectively mapping vertices so that every edge maps to an edge.

For $s\ge 1$, the complete balanced $r$-partite $r$-graph

$K_{s,\dots ,s}^{(r)}$

has disjoint vertex parts $V_1,\dots ,V_r$, each of size $s$, and one hyperedge for every transversal choosing exactly one vertex from each part. For $r=2$ this is the ordinary complete bipartite graph $K_{s,s}$.

For an $r$-uniform hypergraph $F$ with an edge, $\mathrm{ex}(n,F)$ denotes the maximum number of hyperedges in an $n$-vertex $F$-free $r$-uniform hypergraph. The edgeless $r$-graph is an admissible candidate, and the family of possible edge sets is finite, so the maximum exists. The uniformity is determined by $F$.