If one instead starts with the definition of bipartite as *not containing any odd cycles*, then there are results for hypergraphs in this direction. Indeed we can define a *cycle* in a hypergraph as a sequence of distinct alternating vertices and hyperedges $C:=v_1, E_1, v_2, E_2, \dots, v_n, E_n, v_1, E_1$, where $v_i \in E_i \cap E_{i+1}$ for all $i$ (mod $n$). The *length* of $C$ is $n$. The following is a theorem of Gyarfas, Jacobson, Kezdy and Lehel.

**Theorem**. Let $\mathcal{H}$ be a hypergraph where each hyperedge contains at least 3 vertices. If $\mathcal{H}$ contains no odd length cycles, then
$$
\sum_{e \in E(\mathcal{H})} (|e|-1) \leq 2|V(\mathcal{H})| -2.
$$
Moreover, equality holds if and only if $\mathcal{H}$ is the disjoint union of two identical uniform hypertrees.