Tracing a single user

Let g(n,r)g(n,r) be the maximum possible cardinality of a family FF of subsets of {1,2,…,n}{1,2,…,n} so that given a union of at most rr members of FF, one can identify at least one of these members. The study of this function is motivated by questions in molecular biology. We show that g(n,r)=2Θ(nr), thus solving a problem of Csűrös and Ruszinkó.