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Perfect Hashing and Probability

A set of vectors of length t over an alphabet of size b is called k-separated if for every k distinct vectors there is a coordinate in which they all differ. Let N{t,b,k) denote the largest cardinality of such a fc-separated set of vectors. Thus, N = N(t, b, k) is the maximum size of a domain for which there exists a perfect family of (b, k) hash-functions with t members, that is, a family of t functions mapping a domain of size N into a set of size b so that every subset of k elements of the domain is mapped in a one to one fashion by at least one of the functions. The problem of estimating the function N(t,b,k), which is motivated by the numerous applications of perfect hashing in theoretical computer science, has received a considerable amount of attention. The interesting case is that in which t is much bigger than b (and, of course, b > k). Fredman and Komlos [2], (see also [4]) proved that