(I will state my question in terms of combinatorial discrepancy, but the same could be also asked about measure-theoretic discrepancy as well.)
The combinatorial discrepancy of a family $\mathcal F$ is defined as $$disc(\mathcal F)=\min_{\chi\to \pm 1} \max_{F\in \mathcal F} |\sum_{x\in F} \chi(x)|=\min_S \max_{F\in \mathcal F} |2|F\cap S|-|F||,$$
i.e., we want $S$ to be even in all sets of $\mathcal F$. What happens if instead we have two families, one in which we want many elements of $S$, and another in which we want few? Define $$disc(\mathcal F_1,\mathcal F_2)=\min_S \max(\max_{F_1\in \mathcal F_1} 2|F_1\cap S|-|F_1|;\max_{F_2\in \mathcal F_2} |F_2|-2|F_2\cap S|).$$
Note that $disc(\mathcal F,\mathcal F)=disc(\mathcal F)$. Has this parameter ever been studied? Are there some interesting results that carry through? For example, how many permutations do we need in $\mathcal F_1$ and $\mathcal F_2$ to have a non-constant $disc(\mathcal F_1,\mathcal F_2)$? (For the related question in case of $disc(\mathcal F)$ the answer is $3$, see https://arxiv.org/abs/1104.2922.)
