  In the classical combinatorial (adaptive) group testing problem, one is given
two integers \(d\) and \(n\), where \(0\le d\le n\), and a population of \(n\)
items, exactly \(d\) of which are known to be defective. The question is to
devise an optimal sequential algorithm that, at each step, tests a subset of
the population and determines whether such subset is contaminated (i.e.
contains defective items) or otherwise. The problem is solved only when the
\(d\) defective items are identified. The minimum number of steps that an
optimal sequential algorithm takes in general (i.e. in the worst case) to solve
the problem is denoted by \(M(d, n)\). The computation of \(M(d, n)\) appears
to be very difficult and a general formula is known only for \(d = 1\). We
consider here a variant of the original problem, where the size of the subsets
to be tested is restricted to be a fixed positive integer \(k\). The
corresponding minimum number of tests by a sequential optimal algorithm is
denoted by \(M^{\lbrack k\rbrack}(d, n)\). In this paper we start the
investigation of the function \(M^{\lbrack k\rbrack}(d, n)\).
