Much of the work on Turing degrees may be formulated in terms of the embeddability of certain first-order structures in a structure whose universe is some set of degrees and whose relations, functions, and constants are natural degree-theoretic ones. Thus, for example, we know that if (P, ≤P ) is a partial ordering of cardinality at most א1 which is locally countable—each point has at most coun...

We consider the degree-diameter problem for Cayley graphs of dihedral groups. We find upper and lower bounds on the maximum number of vertices of such a graph with diameter 2 and degree d. We completely determine the asymptotic behaviour of this class of graphs by showing that both limits are asymptotically d/2.

Let G be a finite group and φ : V → W an equivariant morphism of finite dimensional G-modules. We say that φ is faithful if G acts faithfully on φ(V ). The covariant dimension of G is the minimum of the dimension of φ(V ) taken over all faithful φ. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alter...