I am framing a particular combinatorial question using users and files for better understanding.
Let there be a universe of files $F$ = $\{f_1, f_2,\ldots,f_n\}$ and $2k$ users $\{u_1, u_2,\ldots, u_{2k}\}$. Each user $u_i$ has a set of files denoted by $F(u_i) \subseteq F$.
Can the users be divided (whenever possible) into two equal sized group $G_1$ and $G_2$ in polynomial time in $n$, such that $\mathcal{F}(G_1) = \bigcup_{u_i \in G_1} F(u_i)$ and similarly let $\mathcal{F}(G_2) = \bigcup_{u_j \in G_2} F(u_j)$ satisfy the following conditions:
$$\mathcal{F}(G_1)\not\subset \mathcal{F}(G_2) \text{ and } \mathcal{F}(G_2)\not\subset \mathcal{F}(G_1), $$ where $\not \subset$ means "not a strict subset".
EDIT 1: The algorithm must to be polynomial time in $k$ and $n$.
Thanks in advance.