A simple proof is given of the best-known upper bound on the cardinality of a set of vectors of length t over an alphabet of size b, with the property that, for every subset of k vectors, there is a coordinate in which they all differ. This question is motivated by the study of perfect hash functions.

We show that, for any finite set P of points in the plane and for any integer k ≥ 2, there is a finite set R = R(P, k) with the following property: for any k-colouring of R there is a monochromatic set , ⊆ R, such that is combinatorially equivalent to the set P, and the convex hull of P contains no point of R \ . We also consider related questions for colourings of p-element subsets of R (p > 1), and show that these analogues have negative solutions.

It is shown that unless NP collapses to random polynomial time RP, there can be no fully polynomial randomised approximation scheme for the antiferromagnetic version of the Q-state Potts model.

In group testing, sets of data undergo tests that reveal if a set contains faulty data. Assuming that data items are faulty with given probability and independently of one another, we investigate small families of tests that enable us to locate correctly all faulty data with probability converging to one as the amount of data grows. Upper and lower bounds on the minimum number of such tests are established for different probability functions, and respective location strategies are constructed.

Consider an electrical network on n nodes with resistors rij between nodes i and j. Let Rij denote the effective resistance between the nodes. Then Foster's Theorem [5] asserts that

where i ∼ j denotes i and j are connected by a finite rij. In [10] this theorem is proved by making use of random walks. The classical connection between electrical networks and reversible random walks implies a corresponding statement for reversible Markov chains. In this paper we prove an elementary identity for ergodic Markov chains, and show that this yields Foster's theorem when the chain is time-reversible.

We also prove a generalization of a resistive inverse identity. This identity was known for resistive networks, but we prove a more general identity for ergodic Markov chains. We show that time-reversibility, once again, yields the known identity. Among other results, this identity also yields an alternative characterization of reversibility of Markov chains (see Remarks 1 and 2 below). This characterization, when interpreted in terms of electrical currents, implies the reciprocity theorem in single-source resistive networks, thus allowing us to establish the equivalence of reversibility in Markov chains and reciprocity in electrical networks.