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Moran, Naor, and Segev have asked what is the minimal r = r(n, k) for which there exists an (n, k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2, ..., n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call [alpha]-fraction k-multiuser tracing [sub.*]((k, [alpha])-FUT (fraction user-tracing) families). We show that r(n, k) = [THETA](k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, [alpha])-FUT families and by constructing an (n, k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant. Index Term--Monotone encoding, multiuser tracing, superimposed codes.