Computing frequent itemsets is one of the most prominent problems in data mining. Recently, a new related problem, called FREQSAT, was introduced and studied: given some itemset–interval pairs, does there exist a database such that for every pair, the frequency of the itemset falls in the interval? In this paper, we extend this FREQSAT-problem by further constraining the database by giving other characteristics as part of the input as well. These characteristics are the maximal transaction length, the maximal number of transactions, and the maximal number of duplicates of a transaction. These extensions and all their combinations are studied in depth, and a hierarchy w.r.t. complexity is given. To make a complete picture, also the cases where the characteristics are constant; i.e., bounded and the bound being a fixed constant that is not a part of the input, are studied.