Rosenthal's lemma in its most basic form states that given an infinite matrix \((m_n)_\) of non-negative reals such that \(\sum_m_n\le 1\) for every \(k\in\omega\), and \(\varepsilon>0\), there exists an infinite set \(A\subset\omega\) such that \(\sum_m_n\le\varepsilon\) for every \(k\in A\). The lemma has numerous important applications in Banach space theory and vector measure theory — I will mention some of them during the talk (on the fly explaining and exemplifying all notions and terms).

A natural question arises — can the choice of a set \(A\) in Rosenthal's lemma be somehow controlled, i.e. can \(A\) be chosen from some fixed family \(\mathcalF\subset[\omega]\omega\)? I will show that it is not possible if \(\mathcalF\) has cardinality strictly less than \(\text(\mathcalM)\) (the covering of category). On the other hand, if \(\mathcalF\) is a basis of a selective ultrafilter (assuming one exists), then \(A\) can be chosen from \(\mathcalF