How many random questions are necessary to identify n distinct objects?

AbstractSuppose that X and A are two finite sets of the same cardinality n ⩾ 2. Assume that there is a bijective mapping φ: X → A which is unknown to us, and we must determine it. We are allowed to ask a sequence of questions each posed as follows. For a given B ⊂ A what is φ−1(B)? In this paper we study a case when the subsets B are chosen uniformly at random. The main result is: if each subset has to split all the atoms of a field generated by the previous subsets, then the total number of questions (needed to determine the mapping completely) is log2 n + (1 + op(1))(2 log2 n)12. Here op(1) stands for a random term approaching 0 in probability as n → ∞