The theoretical upper bounds for duplicate detection in a set of objects?
I think you're asking about the "Element Distinctness Problem."

Assuming you're talking about time complexity (number of comparison operations) rather than space complexity, "lower bound" instead of "upper bound," and that you can't hash elements of your multiset (which gives you $O(n)$ expected time), then the best you can do is $\Theta(n \log n)$.