multi-party Communication complexity of "Set Partition problem" 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.
A linear lower bound on deterministic CC follows by fixing one of the sets to be empty.

For a randomized logarithmic upper bound, first note that this problem can be reduced to the problem asking whether the sum of three $3n$-bit numbers is exactly $2^{3n}-1$.  This one can be solved in $O(\log n)$ randomized communication by the players operating mod a random $O(\log n)$-bit prime.