For a given index set I, let us consider a family (Aν,: ν ∈ I) of subsets of a set E. In this note we deal with some aspects of the following question: to what extent is it possible to prescribe the cardinalities, or the order types in case E is ordered, of the sets Aν and of their pairwise intersections? In (1) the authors have shown that, given any regular cardinal a, there is a family of a+ sets of cardinal a whose pairwise intersections are arbitrarily prescribed to be either less than or equal to a. In Theorem 1 below we prove a stronger result which states that if a is regular, say a = ℵα, and if E is well-ordered and of order type , then one can find a+ subsets Aν, of E, each of type , whose pairwise intersections are arbitrarily prescribed to be either of type ωα or of a type less than ωα. By way of contrast, Theorem 2 below implies – this is its special case m = ℵω; n = ℵ2; p = ℵ0 – that, assuming the Generalized Continuum Hypothesis (GCH), there do not exist ℵω+1 sets Aν, each of cardinal at most ℵω such that ℵ2 of them have pairwise finite intersections, whereas all other pairs of sets Aν have a denumerable intersection. Theorem 3 gives another case in which some type of prescription of the sizes of the intersections cannot be satisfied. Finally, Theorem 4 asserts that in Theorem 3 the condition cfp ≠ cfm cannot be omitted. The paper concludes with some remarks on open questions.