I was asked to post a different question following a wording error on [my previous question](https://mathoverflow.net/questions/442533/jigsaw-puzzle-on-set-family), so here it is.



> 
> Given a set family $\mathcal{F}$ of $[n]$ (with certain additional properties), such that every element of $[n]$ lies in exactly $K$ sets in $\mathcal{F}$, what properties of $\mathcal{F}$ are sufficient to guarantee that we can partition $\mathcal{F}$ into $K$ subfamilies $\mathcal{F}\_i$ such that each subfamily is a partition of $[n]$?
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What I am looking for is a nontrivial sufficient condition that guarantees the existence of such a partition of $\mathcal{F}$(e.g. in the style of Hall's theorem), and whether any papers have studied these kinds of questions before.


I feel this kind of question is very natural, and greatly appreciate any reference on this type of question.



"JigSaw Puzzle" on Set Family II
This problem is NP-complete, a nice reference is this answer here by András Salamon:
<https://cstheory.stackexchange.com/a/356/419>


If you are interested in results about the complete family, see [Baranyai's theorem](https://en.wikipedia.org/wiki/Baranyai%27s_theorem).


