 
Summary: Tracing Many Users With Almost No Rate Penalty
Noga Alon
Vera Asodi
Abstract
For integers n, r 2 and 1 k r, a family F of subsets of [n] = {1, . . . , n} is called kout
ofr multiple user tracing if, given the union of any r sets from the family, one can identify at
least min(k, ) 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 paper 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 , c2
k2 ) log h(n,r,k)
n min(c3
r , c4 log k
k2 ). In particular, for all k
r, log h(n,r,k)
n = (1/r).
This improves an estimate of Laczay and Ruszink´o.
