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What is the maximal number of sets in a set system $\mathcal{A}$ of subsets of an $n$ element set such that or every $i \neq j $ and $A_i,A_j \in \mathcal{A}$ the difference $A_i \setminus A_j$ is unique? (No other sets produce the same difference!)

What is the maximal number of sets in a set system $\mathcal{A}$ of subsets of an $n$ element set such that for every $i \neq j $ and $A_i,A_j \in \mathcal{A}$ the difference $A_i \setminus A_j$ is unique? (No other sets produce the same difference!)