This question attempts to generalize Hall's theorem to taking "triangles" instead of edges.


Say we are given $3$ disjoint sets over vertices $A,B,C$. Consider a graph who's edges only connect vertices from those $3$ different sets. 
We say this arrangment is $A$-good if there exists $|A|$ disjoint triangles in this graph.


Is there a equivalent condition to being $A$-good similiar to the one in Hall's theorem?



Generlization of Halls theorem to triangles
Determining whether a $k$-partite $k$-uniform hypergraph has a perfect matching is NP-complete for $k\geq 3$, so any condition equivalent to being $A$-good is bound to be somewhat complicated. Finding a neccessary and sufficient conditions for hypergraphs is considered a difficult open problem for which a lot of partial progress has been done.


With that said, there is a generalization of Hall's theorem to multipartite hypergraphs in ["Hall's Theorem for Hypergraphs"](https://onlinelibrary.wiley.com/doi/abs/10.1002/1097-0118%28200010%2935%3A2%3C83%3A%3AAID-JGT2%3E3.0.CO%3B2-V) by Aharoni and Haxell which gives a sufficient condition for perfect matchings in hypergraphs.


I'll describe the theorem here for the 3-partite case that you are interested in. Given a graph $G$ and a matching $M$ we define $f(M)$ to be the size of the smallest matching which intersects every edge of $M$. Next we define the *matching width* of $G$ as $\operatorname{mw}(G)=\max\_{M} f(M)$.



> 
> **Theorem (Aharoni-Haxell)**: Suppose $A=\{a\_1,a\_2,\dots a\_n\}$. Let $G\_i$ be the graph with vertex set $B\cup C$ whose edges are those pairs $\{b,c\}$ for which $a\_ib,a\_ic,bc$ are all edges of the original graph. If the condition
>  $$\operatorname{mw}\left(\bigcup\_{i\in I}G\_i\right)\geq |I|$$
>  is satisfied for all $I\subset A$ then your graph is $A$-good.
> 
> 
> 


[Here](https://gilkalai.wordpress.com/2012/11/25/happy-birthday-ron-aharoni/) is a blog post by Gil Kalai which reviews some results and applications in this area. There are similar results that can be phrased by using other graph parameters than the matching width, but they all have in common the fact that the proofs are topological in nature.


