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We study inhomogeneous random graphs in the subcritical case. Among other results, we derive an exact formula for the size of the largest connected component scaled by log$n$, with$n$being the size of the graph. This generalizes a result for the “rank-1 case”. We also investigate branching processes associated with these graphs. In particular, we discover that the same well-known equation for the survival probability, whose positive solution determines the asymptotics of the size of the largest component in the supercritical case, also plays a crucial role in the subcritical case. However, now it is the$negative$solutions that come into play. We disclose their relationship to the distribution of the progeny of the branching process.

We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by$δ$are called$δ$-$uniform$. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.

A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at$m$∼$n$/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.

Let each point of a homogeneous Poisson process in ℝ^{$d$}independently be equipped with a random number of stubs (half-edges) according to a given probability distribution$μ$on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution$μ$. Leaving aside degenerate cases, we prove that for any$μ$there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale–Shapley stable marriage, we give sufficient conditions on$μ$for the absence and presence of infinite components.

In a uniform random recursive$k$-directed acyclic graph, there is a root, 0, and each node in turn, from 1 to$n$, chooses$k$uniform random parents from among the nodes of smaller index. If$S$_{$n$}is the shortest path distance from node$n$to the root, then we determine the constant$σ$such that$S$_{$n$}/log$n$→$σ$in probability as$n$→∞. We also show that max_{1≤$i$≤$n$}$S$_{$i$}/log$n$→$σ$in probability.

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph ℋ, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.