Tracing many users with almost no rate penalty

For integers n, r ≥ 2 and 1 ≤ k ≤ r, a family F of subsets of [n] = {1,⋯,n} is called k-out-of-r multiple-user tracing if, given the union of any ℓ ≤ r sets from the family, one can identify at least (min(k, ℓ) 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) ≤ log h(n, r, k)/n ≤ min (c₂/r, c₄ log k/k²). In particular, for all k ≤ √r, log h(n, r, k)/n = θ(1/r). This improves an estimate of Laczay and Ruszinkó.