A uniform random intersection graph G(n, m, k) is a random graph constructed as follows. Label each of n nodes by a randomly chosen set of k distinct colours taken from some finite set of possible colours of size m. Nodes are joined by an edge if and only if some colour appears in both their labels. These graphs arise in the study of the security of wireless sensor networks, in particular when modelling the network graph of the well known key predistribution technique due to Eschenauer and Gligor.The paper determines the threshold for connectivity of the graph G(n, m, k) when n → ∞ in many situations. For example, when k is a function of n such that k ≥ 2 and m = ⌊n α ⌋ for some fixed positive real number α then G(n, m, k) is almost surely connected when lim inf k 2 n/m log n > 1, and G(n, m, k) is almost surely disconnected when lim sup k 2 n/m log n < 1.

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of 2 n−1 permutations that pairwise generate the symmetric group S n . There is no set of 2 n−1 + 1 permutations having this property. For all sufficiently large integers n with n ≡ 2 mod 4, there exists a set of 2 n−2 even permutations that pairwise generate the alternating group A n . There is no set of 2 n−2 + 1 permutations having this property.

No abstract

How many groups of order n are there? This is a natural question for anyone studying group theory, and this Tract provides an exhaustive and up-to-date account of research into this question spanning almost fifty years. The authors presuppose an undergraduate knowledge of group theory, up to and including Sylow's Theorems, a little knowledge of how a group may be presented by generators and relations, a very little representation theory from the perspective of module theory, and a very little cohomology theory - but most of the basics are expounded here and the book is more or less self-contained. Although it is principally devoted to a connected exposition of an agreeable theory, the book does also contain some material that has not hitherto been published. It is designed to be used as a graduate text but also as a handbook for established research workers in group theory.

Let V be a set of order n and let F be a set of order q. A set S [,: V Ä F ] of functions from V to F is an (n, q, t)-perfect hash family if for all X V with |X | =t, there exists , # S which is injective when restricted to X. Perfect hash families arise in compiler design, in circuit complexity theory and in cryptography. Let S be an (n, q, t)-perfect hash family. The paper provides lower bounds on |S|, which better previously known lower bounds for many parameter sets. The paper exhibits new classes of perfect hash families which show that these lower bounds are realistic. Academic Press