Bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of
"activation" on a given realization of the graph with a given number of
initially active nodes. At each step those vertices which have not been active
but have at least $r\geq 2$ active neighbours become active as well. We study
the size $A^*$ of the final active set. The parameters of the model are,
besides $r$ (fixed) and $n$ (tending to $\infty$), the size $a=a(n)$ of the
initially active set and the probability $p=p(n)$ of the edges in the graph.

We study the maximum of a Brownian motion with a parabolic drift; this is a
random variable that often occurs as a limit of the maximum of discrete
processes whose expectations have a maximum at an interior point. We give
series expansions and integral formulas for the distribution and the first two
moments, together with numerical values to high precision.

We study the susceptibility, i.e., the mean cluster size, in random graphs
with given vertex degrees. We show, under weak assumptions, that the
susceptibility converges to the expected cluster size in the corresponding
branching process. In the supercritical case, a corresponding result holds for
the modified susceptibility ignoring the giant component and the expected size
of a finite cluster in the branching process; this is proved using a duality
theorem.

In this paper we study the minimal number of translates of an arbitrary
subset $S$ of a group $G$ needed to cover the group, and related notions of the
efficiency of such coverings. We focus mainly on finite subsets in discrete
groups, reviewing the classical results in this area, and generalizing them to
a much broader context. For example, we show that while the worst-case
efficiency when $S$ has $k$ elements is of order $1/\log k$, for $k$ fixed and
$n$ large, almost every $k$-subset of any given $n$-element group covers $G$
with close to optimal efficiency.

We study the limit theory of large threshold graphs and apply this to a
variety of models for random threshold graphs. The results give a nice set of
examples for the emerging theory of graph limits.