We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...) c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A\B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of "computable at density r" where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness. (shrink)

We introduce agreement reducibility and highlight its major features. Given subsets A and B of, we write if there is a total computable function satisfying for all,.We shall discuss the central role plays in this reducibility and its connection to strong‐hyper‐hyper‐immunity. We shall also compare agreement reducibility to other well‐known reducibilities, in particular s1‐ and s‐reducibility. We came upon this reducibility while studying the computable reducibility of a class of equivalence relations on based on set‐agreement. We end by describing the (...) origin of agreement reducibility and presenting some results in that context. (shrink)