Roughly speaking, a pattern is a finite sequence coding the set of natural numbers n for which the Σ n + 1 projectum is less than the Σ n projectum for a given admissible ordinal. We prove that for each pattern there exists an ordinal realizing it. Several results on the orderings of patterns are given. We conclude the paper with remarks on ▵ n projecta. The main technique, used throughout the paper, is Jensen's Uniformisation Theorem.

We present several examples of hereditary classes of finite structures satisfying the joint embedding property and the weak amalgamation property, but failing the cofinal amalgamation property. These include a continuum‐sized family of classes of finite undirected graphs, as well as an example due to Pouzet with countably categorical generic limit.

We make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.