Yahya ould Hamidoune passed away in Paris on 11 March 2011 after a brief illness, leaving insufficient time for his friends and colleagues to express their indebtedness to him for his kindness and generosity, both in mathematics and in everyday life. Yahya was a discreet individual, always looking for the essential rather than the superficial, and certainly did not receive the recognition he deserved. May this modest testimony render justice to this singular man.

Philippe Flajolet, mathematician and computer scientist extraordinaire, the father of analytic combinatorics, suddenly passed away on 22 March 2011, at the prime of his career. He is celebrated for opening new lines of research in the analysis of algorithms, developing powerful new methods, and solving difficult open problems. His research contributions will have an impact for generations, and his approach to research, based on curiosity, discriminating taste, broad knowledge and interests, intellectual integrity, and a genuine sense of camaraderie, will serve as an inspiration to those who knew him, for years to come.

We offer a unified approach to the theory of concave majorants of random walks, by providing a path transformation for a walk of finite length that leaves the law of the walk unchanged whilst providing complete information about the concave majorant. This leads to a description of a walk of random geometric length as a Poisson point process of excursions away from its concave majorant, which is then used to find a complete description of the concave majorant of a walk of infinite length. In the case where subsets of increments may have the same arithmetic mean, we investigate three nested compositions that naturally arise from our construction of the concave majorant.

In this paper we explore first passage percolation (FPP) on the Erdős–Rényi random graph Gn(pn), where we assign independent random weights, having an exponential distribution with rate 1, to the edges. In the sparse regime, i.e., when npn → λ > 1, we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to (λ log n)/(λ − 1). Furthermore, we prove that the minimal weight centred by (log n)/(λ − 1) converges in distribution.

We also investigate the dense regime, where npn → ∞. We find that although the base graph is ultra-small (meaning that graph distances between uniformly chosen vertices are o(log n)), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount Hn satisfies the universality property that whatever the value of pn, Hn/log n → 1 in probability and, more precisely, (Hn − βn log n)/, where βn = λn/(λn − 1), has a limiting standard normal distribution. The constant βn can be replaced by 1 precisely when λn ≫ , a case that has appeared in the literature (under stronger conditions on λn) in [4, 13]. We also find lower bounds for the maximum, over all pairs of vertices, of the optimal weight and hopcount.

This paper continues the investigation of FPP initiated in [4] and [5]. Compared to the setting on the configuration model studied in [5], the proofs presented here are much simpler due to a direct relation between FPP on the Erdős–Rényi random graph and thinned continuous-time branching processes.

An old result by Shearer relates the Lovász local lemma with the independent set polynomial on graphs, and consequently, as observed by Scott and Sokal, with the partition function of the hard-core lattice gas on graphs. We use this connection and a recent result on the analyticity of the logarithm of the partition function of the abstract polymer gas to get an improved version of the Lovász local lemma. As an application we obtain tighter bounds on conditions for the existence of Latin transversal matrices.

Let Δ ≥ 3 be an integer. Given a fixed z ∈ +Δ such that zΔ > 0, we consider a graph Gz drawn uniformly at random from the collection of graphs with zin vertices of degree i for i = 1,. . .,Δ. We study the performance of the Karp–Sipser algorithm when applied to Gz. If there is an index δ > 1 such that z1 = . . . = zδ−1 = 0 and δzδ,. . .,ΔzΔ is a log-concave sequence of positive reals, then with high probability the Karp–Sipser algorithm succeeds in finding a matching with n ∥ z ∥ 1/2 − o(n1−ε) edges in Gz, where ε = ε (Δ, z) is a constant.

Our aim in this note is to make some conjectures about extremal densities of daisy-free families, where a ‘daisy’ is a certain hypergraph. These questions turn out to be related to some Turán problems in the hypercube, but they are also natural in their own right. We start by giving the daisy conjectures, and some related problems, and shall then go on to describe the connection with vertex-Turán problems in the hypercube.

Let (r−1)n ≥ rk and let . Suppose that F1 ∩ ⋅⋅⋅ ∩ Fr ≠ ∅ holds for all Fi ∈ i, 1 ≤ i ≤ r. Then we show that .

The tools of zero biasing are adapted to yield a general result suitable for analysing the behaviour of certain growth processes. The main theorem is applied to prove a central limit theorem, with explicit error terms in the L1 metric, for a natural statistic of the Jack measure on partitions.

The classical Erdős–Pósa theorem states that for each positive integer k there is an f(k) such that, in each graph G which does not have k + 1 disjoint cycles, there is a blocker of size at most f(k); that is, a set B of at most f(k) vertices such that G−B has no cycles. We show that, amongst all such graphs on vertex set {1,. . .,n}, all but an exponentially small proportion have a blocker of size k. We also give further properties of a random graph sampled uniformly from this class, concerning uniqueness of the blocker, connectivity, chromatic number and clique number.

A key step in the proof of the main theorem is to show that there must be a blocker as in the Erdős–Pósa theorem with the extra ‘redundancy’ property that B–v is still a blocker for all but at most k vertices v ∈ B.

We give a short proof of the following result on the distribution of three-term arithmetic progressions in sparse subsets of Fpn. For every α > 0 there exists a constant C = C(α) such that the following holds for all r ≥ Cpn/2 and for almost all sets R of size r of Fpn. Let A be any subset of R of size at least αr; then A contains a non-trivial three-term arithmetic progression. This is an analogue of a hard theorem by Kohayakawa, Łuczak and Rödl. The proof uses a version of Green's regularity lemma for subsets of a typical random set, which is of interest in its own right.

A set of reals A = {a1,. . .,an} is called convex if ai+1 − ai > ai − ai−1 for all i. We prove, among other results, that for some c > 0 every convex A satisfies |A−A| ≥ c|A|8/5log−2/5|A|.