Tracing a single user

Let g ( n , r ) be the maximum possible cardinality of a family F of subsets of { 1 , 2 , … , n } so that given a union of at most r members of F , 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 ) = 2 Θ ( n r ) , thus solving a problem of Csűrös and Ruszinkó.