Extremal Uniquely Resolvable Multisets

Varun Sivashankar

Abstract: For positive integers and , consider a multiset of nonempty subsets of such that there is a unique partition of these subsets into partitions of . We study the maximum possible size of such a multiset. We focus on the regime and show that . When for any , this lower bound simplifies to , and we show a matching upper bound that is optimal up to a factor of . We also compute exactly when .