Uncountable constructions for B.A. e.c. groups and banach spaces

This paper has two aims: to aid a non-logician to construct uncountable examples by reducing the problems to finitary problems, and also to present some construction solving open problems. We assume the diamond for א ₁ and solve problems in Boolean algebras, existentially closed groups and Banach spaces. In particular, we show that for a given countable e.c. group M there is no uncountable group embeddable in every G {Mathematical expression}-equivalent to M; and that there is a non-separable Banach space with no א ₁ elements, no one being the closure of the convex hull of the others. Both had been well-known questions. We also deal generally with inevitable models (§4).