Birkhoff's theorem says that, in a bipartite graph $G$ in which both sides have size $n$, any fractional matching of size $n$ can be presented as a convex combination of integral matchings of size $n$ (at most $n^2-2 n +2$ such matchings are needed).
Consider a tripartite hypergraph $H$ in which all three sides have size $n$. In contrast to the bipartite case, a fractional matching of size $n$ does not imply the existence of an integral matching of size $n$, so a direct generalization of Birkhoff's algorithm is not true. Is there a weaker theorem that holds in the case of tripartite hypergraphs? For example, is it true that a fractional matching of size $n$ can be decomposed into integral matchings of size smaller than $n$?
