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.

In this paper we prove that given a finite collection of finite graphs, and the subsets of vertices of a random graph G that induce those graphs, it is almost always possible to uniquely reconstruct a class of graphs equivalent to G.

Analogues of the Erdős-Ko-Rado theorem are proved for the Boolean algebra of all subsets of {1,…n} and in this algebra truncated by the removal of the empty set and the whole set.

We show by elementary methods that given any finite partition of the set ℕ of positive integers, there is one cell that is both additively and multiplicatively rich. In particular, this cell must contain a sequence and all of its finite sums, and another sequence and all of its finite products, a fact that was previously known only by utilizing the algebraic structure of the Stone–Čech compactification βℕ of ℕ.

Notions of deletion and contraction for the class of set functions from finite sets into the integers are defined. An operation on a subclass of such set functions is a function from the subclass into itself that preserves ground sets and respects isomorphism. The operations on set functions that interchange deletion and contraction are characterised, as are those with the further property of being involutary. Similar results are given for polymatroids. There is a unique involutary operation on the class of k-polymatroids that interchanges deletion and contraction. The results generalise those of Kung [3].

A polynomial-time randomised algorithm for uniformly generating forests in a dense graph is presented. Using this, a fully polynomial randomised approximation scheme (fpras) for counting the number of forests in a dense graph is created.

The partition number of a product hypergraph is introduced as the minimal size of a partition of its vertex set into sets that are edges. This number is shown to be multiplicative if all factors are graphs with all loops included.

Jackson [10] gave a polynomial sufficient condition for a bipartite tournament to contain a cycle of a given length. The question arises as to whether deciding on the maximum length of a cycle in a bipartite tournament is polynomial. The problem was considered by Manoussakis [12] in the slightly more general setting of 2-edge coloured complete graphs: is it polynomial to find a longest alternating cycle in such coloured graphs? In this paper, strong evidence is given that such an algorithm exists. In fact, using a reduction to the well known exact matching problem, we prove that the problem is random polynomial.

Dowling lattices are a class of geometric lattices, based on groups, which have been shown to share many properties with projective geometries. In this paper we show that the automorphisms of Dowling lattices are analogs of the automorphisms of projective geometries. We also treat similar results for several related geometric lattices.

A harmonious colouring of a simple graph G is a proper vertex colouring such that each pair of colours appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colours in such a colouring.

For any positive integer m, let Q(m) be the least positive integer k such that ≥ m. We show that for almost all unlabelled, unrooted trees T, h(T) = Q(m), where m is the number of edges of T.

We consider a random digraph Din, out(n) on vertices 1, …, n, where, for each vertex v, we choose at random one of the n possible arcs with head v and one of the n possible arcs with tail v. We show that the expected size of the largest component of Din, out is .

We give some sufficient conditions for an (S, U)-outline T-factorization of Kn to be an (S, U)-amalgamated T-factorization of Kn. We then apply these to give various necessary and sufficient conditions for edge coloured graphs G to have recoverable embeddings in T-factorized Kn's.

The study of asymptotics of random permutations was initiated by Erdős and Turáan, in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more group-theoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions).

This paper deals with infinite binary sequences. Each sequence is treated as generated by a nondeterministic shift register. A measure-theoretic criterion helpful in finding a deterministic generator of the set of sequences is proposed.

A graph may be regarded as an electrical network in which each edge has unit resistance. We obtain explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case. A well-known link with random walks motivates a conjecture about the maximum effective resistance. Arguments are given that point to the truth of the conjecture for all known distance-regular graphs.

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.