  We examine the following version of a classic combinatorial search problem
introduced by R\'enyi: Given a finite set $X$ of $n$ elements we want to
identify an unknown subset $Y \subset X$ of exactly $d$ elements by testing, by
as few as possible subsets $A$ of $X$, whether $A$ contains an element of $Y$
or not. We are primarily concerned with the model where the family of test sets
is specified in advance (non-adaptive) and each test set is of size at most a
given $k$. Our main results are asymptotically sharp bounds on the minimum
number of tests necessary for fixed $d$ and $k$ and for $n$ tending to
infinity.
