Consider the following problem:


*Let $F \subseteq 2^{I}$ be a finite family of finite subsets of some index set $I.$*


*Let $F\_x$ be defined as the number of elements of $F$ that contains $x.$*


*Assume that for each $x \in I,$ $F\_x$ is an even number.
Under what conditions on $F$ can we find a subset $F'$ of $F$ such that
$2F'\_x = F\_x$ for all $x$?*


Note that if $I$ is the set of natural numbers, then for any family $F$ which consists of intervals (satisfying the extra property that $F\_x$ is always even), this problem is solvable, and it is easy to prove.
(However, it does not generalize to rectangles, where there are counter examples)


Now, there are several ways to generalize/specialize this problem in various directions, for example, [this](https://mathoverflow.net/questions/132651/integer-solution-to-special-system-of-linear-equations) question is such a generalization.


*I am curios if there is any research done on problems that is similar to this type of problem above.*



Overlapping sets
Your question is equivalent to whether the combinatorial discrepancy of the dual hypergraph is zero. (In case you are unfamiliar with the notion, start here: <http://en.wikipedia.org/wiki/Discrepancy_of_hypergraphs>) This problem is well-known to be NP-complete, you can easily reduce e.g., 1-in-3 SAT to it.


