TITLE: Is there a version of König's theorem for tripartite 3-graphs?
QUESTION [13 upvotes]: 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!

REPLY [18 votes]: 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 
Ryser's conjecture for tripartite 3-graphs.  Combinatorica 21 (2001), no. 1, 1--4, 
and you can find a copy of the paper here.