In an application I'm considering, I need to know the communication complexity of the following problem:
Given $n$, let $S$ be the set of integers from $1$ to $n$. Alice, Bob, and Carol each receives a subset of $S$, denoted by $A$, $B$, and $C$, respectively. They want to check whether $A$, $B$ and $C$ form a partition of $S$, i.e., they are disjoint and their union is $S$.
I'm particularly interested in the case of 3 parties but other cases would be interesting as well. Note that for the case of 2 parties, the problem is equivalent to EQUALITY problem so it has $\Omega(n)$ lower bound for deterministic protocols but $O(\log n)$ upper bound for randomized protocols.
My question is whether this problem is known before. If you know any problems that might be related, I would be interested to know as well.