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

In other words, let G = (V,T) be a tripartite graph, with V set of vertices ($V$ union of three disjoint subsets $A,B,C$) and T set of triples $\{v_1,v_2,v_3\}$, with $v_1 \in A, v_2 \in B, v_3 \in C$. We say maximum 3 - matching the maximum number of triples such that for any pair of them there are no vertices in common.

Any suggestions or references? Thanks in advance!

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Check out Ryser's conjecture here. It looks like you are interested in the case r=3 (3-uniform, 3-partite hypergraphs), which was recently settled by Aharoni. The reference is

Ron Ryser's conjecture for tripartite 3-graphs. Combinatorica 21 (2001), no. 1, 1--4.

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