Let [Mscr]n,k(S) be the set of n-edge k-vertex rooted maps in some class on the surface S. Let P be a planar map in the class. We develop a method for showing that almost all maps in [Mscr]n,k(S) contain many copies of P. One consequence of this is that almost all maps in [Mscr]n,k(S) have no symmetries. The classes considered include c-connected maps (c [les ] 3) and certain families of degree restricted maps.

A tournament T on a set V of n players is an orientation of the edges of the complete graph Kn on V; T will be called a random tournament if the directions of these edges are determined by a sequence {Yj[ratio ]j = 1, …, (n2)} of independent coin flips. If (y, x) is an edge in a (random) tournament, we say that y beats x. A set A ⊂ V, |A| = k, is said to be beaten if there exists a player y ∉ A such that y beats x for each x ∈ A. If such a y does not exist, we say that A is unbeaten. A (random) tournament on V is said to have property Sk if each k-element subset of V is beaten. In this paper, we use the Stein–Chen method to show that the probability distribution of the number W0 of unbeaten k-subsets of V can be well-approximated by that of a Poisson random variable with the same mean; an improved condition for the existence of tournaments with property Sk is derived as a corollary. A multivariate version of this result is proved next: with Wj representing the number of k-subsets that are beaten by precisely j external vertices, j = 0, 1, …, b, it is shown that the joint distribution of (W0, W1, …, Wb) can be approximated by a multidimensional Poisson vector with independent components, provided that b is not too large.

Assemblies are labelled combinatorial objects that can be decomposed into components. Examples of assemblies include set partitions, permutations and random mappings. In addition, a distribution from population genetics called the Ewens sampling formula may be treated as an assembly. Each assembly has a size n, and the sum of the sizes of the components sums to n. When the uniform distribution is put on all assemblies of size n, the process of component counts is equal in distribution to a process of independent Poisson variables Zi conditioned on the event that a weighted sum of the independent variables is equal to n. Logarithmic assemblies are assemblies characterized by some θ > 0 for which i[]Zi → θ. Permutations and random mappings are logarithmic assemblies; set partitions are not a logarithmic assembly. Suppose b = b(n) is a sequence of positive integers for which b/n → β ε (0, 1]. For logarithmic assemblies, the total variation distance db(n) between the laws of the first b coordinates of the component counting process and of the first b coordinates of the independent processes converges to a constant H(β). An explicit formula for H(β) is given for β ε (0, 1] in terms of a limit process which depends only on the parameter θ. Also, it is shown that db(n) → 0 if and only if b/n → 0, generalizing results of Arratia, Barbour and Tavaré for the Ewens sampling formula. Local limit theorems for weighted sums of the Zi are used to prove these results.

A model for a random random-walk on a finite group is developed where the group elements that generate the random-walk are chosen uniformly and with replacement from the group. When the group is the d-cube Zd2, it is shown that if the generating set is size k then as d → ∞ with k − d → ∞ almost all of the random-walks converge to uniform in k ln (k/(k − d))/4+ρk steps, where ρ is any constant satisfying ρ > −ln (ln 2)/4.

Let G be a loopless 2-connected graph whose largest cycle has c edges, and whose largest bond has c* edges. We show that |E(G)| [les ] ½cc*.

An [n, k, r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, V1, …, Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 [les ] i < j [les ] n. An independent transversal in an [n, k, r]-partite graph is an independent set, T, consisting of n vertices, one from each Vi. An independent covering is a set of k pairwise-disjoint independent transversals. Let t(k, r) denote the maximal n for which every [n, k, r]-partite graph contains an independent transversal. Let c(k, r) be the maximal n for which every [n, k, r]-partite graph contains an independent covering. We give upper and lower bounds for these parameters. Furthermore, our bounds are constructive. These results improve and generalize previous results of Erdo″s, Gyárfás and Łuczak [5], for the case of graphs.

Lemke and Kleitman [2] showed that, given a positive integer d and d (necessarily non-distinct) divisors of da1, …, ad there exists a subset Q ⊆ {1, …, d} such that d = [sum ]i∈Qai answering a conjecture of Erdo″s and Lemke. Here we extend this result, showing that, provided [sum ]p|d1/p [les ] 1 (where the sum is taken over all primes p), there is some collection from a1, …, ad which both sum to d and which can themselves be ordered so that each element divides its successor in the order. Furthermore, we shall show that the condition on the prime divisors is in some sense also necessary.