The λ-dilate of a set A is λċA={λa : a∈A}. We give an asymptotically sharp lower bound on the size of sumsets of the form λ1ċA+ċċċ+λkċA for arbitrary integers λ1,. . .,λk and integer sets A. We also establish an upper bound for such sums, which is similar to, but often stronger than Plünnecke's inequality.

We consider the problem of minimizing the size of a family of sets such that every subset of {1,. . ., n} can be written as a disjoint union of at most k members of , where k and n are given numbers. This problem originates in a real-world application aiming at the diversity of industrial production. At the same time, the question of finding the minimum of || so that every subset of {1,. . ., n} is the union of two sets in was asked by Erdős and studied recently by Füredi and Katona without requiring the disjointness of the sets. A simple construction providing a feasible solution is conjectured to be optimal for this problem for all values of n and k and regardless of the disjointness requirement; we prove this conjecture in special cases including all (n, k) for which n≤3k holds, and some individual values of n and k.

We study the number of subtrees on the fringe of random recursive trees and random binary search trees whose limit law is known to be either normal or Poisson or degenerate depending on the size of the subtree. We introduce a new approach to this problem which helps us to further clarify this phenomenon. More precisely, we derive optimal Berry–Esseen bounds and local limit theorems for the normal range and prove a Poisson approximation result as the subtree size tends to infinity.

Let n ≥ 1 be an integer. Given a vector a=(a1,. . ,an)∈, write(the ‘projection of a onto the positive orthant’). For a set A⊆ put A+:={a+: a ∈ A} and A−A:={a−b: a, b ∈ A}. Improving previously known bounds, we show that |(A−A)+| ≥ |A|3/5/6 for any finite set A⊆, and that |(A−A)+| ≥ c|A|6/11/(log |A|)2/11 with an absolute constant c>0 for any finite set A⊆ such that |A| ≥ 2.

We show that for each α>0 every sufficiently large oriented graph G with δ+(G), δ−(G)≥3|G|/8+α|G| contains a Hamilton cycle. This gives an approximate solution to a problem of Thomassen [21]. In fact, we prove the stronger result that G is still Hamiltonian if δ(G)+δ+(G)+δ−(G)≥3|G|/2 + α|G|. Up to the term α|G|, this confirms a conjecture of Häggkvist [10]. We also prove an Ore-type theorem for oriented graphs.

The number of spanning trees in the giant component of the random graph (n, c/n) (c > 1) grows like exp{m(f(c)+o(1))} as n → ∞, where m is the number of vertices in the giant component. The function f is not known explicitly, but we show that it is strictly increasing and infinitely differentiable. Moreover, we give an explicit lower bound on f′(c). A key lemma is the following. Let PGW(λ) denote a Galton–Watson tree having Poisson offspring distribution with parameter λ. Suppose that λ*>λ>1. We show that PGW(λ*) conditioned to survive forever stochastically dominates PGW(λ) conditioned to survive forever.