# Is there a version of König's theorem for tripartite 3-graphs?

I would like to know if there exists a version of König's theorem for tripartite $3$-graphs.

In other words, let $G = (V,T)$ be a tripartite $3$-graph. That is, $V$ is a set of vertices (with $V$ equal to the union of three disjoint subsets $A,B,C$) and $T$ is a set of hyperedges $\{v_1,v_2,v_3\}$, with $v_1 \in A, v_2 \in B, v_3 \in C$. A matching is a collection of hyperedges such that no two of them share a vertex. A cover is a set of vertices that meets every hyperedge.

Is there a relationship between the size of a maximum matching and the size of a minimum cover in $G$.

Any suggestions or references? Thanks in advance!

This is a special case of Ryser's Conjecture, which states that in an $r$-partite, $r$-uniform hypergraph (with $r>1$)
$\tau \leq (r-1) \nu$,
where $\tau$ is the size of a minimum cover and $\nu$ is the size of a maximum matching. Note that the case $r=2$ is simply König's theorem.
You are interested in the case $r=3$, which was settled by Aharoni. The reference is