Tracing Many Users With Almost No Rate Penalty

For integers n,rges2 and 1lesklesr, a family F of subsets of [n]={1,...,n} is called k- out-of-r multiple-user tracing if, given the union of any lscrlesr sets from the family, one can identify at least min(k,lscr) of them. This is a generalization of superimposed families (k=r) and of single-user tracing families (k=1). The study of such families is motivated by problems in molecular biology and communication. In this correspondence, we study the maximum possible cardinality of such families, denoted by h(n,r,k), and show that there exist absolute constants c₁,c₂,c₃,c₄>0 such that min (c₁/r,c₃/k²)leslog h(n,r,k)/n les min (c₂/r,c₄logk/k²). In particular, for all klesradicr,log h(n,r,k)/n=Theta(1/r). This improves an estimate of Laczay and Ruszinkoacute