On the Existence of Atomic Models

We give an example of a countable theory T such that for every cardinal λ ≥ א2 there is a fully indiscernible set A of power λ such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T (A) is a theory of size λ where the principal types are dense, yet T (A) has no atomic model. If a complete theory T has an atomic model then the principal types are dense in the Stone space Sn(T ) for each n ∈ ω. In [2, Theorem 1.3], [3, page 168] and [5, IV 5.5], Knight, Kueker and Shelah independently showed that the converse holds, provided that the cardinality of the underlying language has size at most א1. Partially supported by an NSF Postdoctoral Fellowship. The authors thank the U.S. Israel Binational Science Foundation for its support of this project. This is item 489 in Shelah’s bibliography.