We consider subgraph counts in general preferential attachment models with power-law degree exponent $\tau > 2$. For all subgraphs H, we find the scaling of the expected number of subgraphs as a power of the number of vertices. We prove our results on the expected number of subgraphs by defining an optimization problem that finds the optimal subgraph structure in terms of the indices of the vertices that together span it and by using the representation of the preferential attachment model as a Pólya urn model.

We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph that does not contain $k\,+\,1$ vertex-disjoint triangles.

This extends a result of Moon [Canad. J. Math. 20 (1968), 96–102], which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corrádi–Hajnal Theorem.

By using an adaptation of the radial generation method, we give an integral formula for the proportion of triangles in a Poisson-Voronoi tessellation, which gives a value of 0.0112354 to 7 decimal places. We also obtain the first four moments of some characteristics of triangles.

The paper considers the bias of Bookstein's mean estimator for shape under the isotropic normal model. This work depends on certain distributional properties of shape variables. An alternative unbiased modified estimator is proposed and its performance is compared with various estimators, including Procrustes and the exact maximum likelihood estimator, in a simulation study.

In this paper we investigate the exact shape distribution for general Gaussian labelled point configurations in two dimensions. The shape density is written in a closed form, in terms of Kendall's or Bookstein's shape variables. The distribution simplifies considerably in certain cases, including the complex normal, isotropic, circular Markov and equal means cases. Various asymptotic properties of the distribution are investigated, including a large variation distribution and the normal approximation for small variations. The triangle case is considered in particular detail, and we compare the density with simulated densities for some examples. Finally, we consider inference problems, with an application in biology.