  This paper is a survey of recent results and methods in (Tarskian) algebraic
logic. We focus on cylindric algebras. Fix 2<n<\omega. Rainbow constructions
are used to solve problems on classes consisting of algebras having a neat
embedding property substantially generalizing seminal results of Hodkinson as
well as Hirsch and Hodkinson on atom-canonicity and complete representations,
respectively. For proving non-atom-canonicity of infinitely many varieties
approximating the variety of representable algebras of dimension n, so-called
blow up and blur constructions are used. Rainbow constructions are compared to
constructions using Monk-like algebras and cases where both constructions work
are given. When splitting methods fail. rainbow constructions are used to show
that diagonal free varieties of representable diagonal free algebras of finite
dimension n, do no admit universal axiomatizations containing only finitely
many variables. Notions of representability, like complete, weak and strong are
lifted from atom structures to atomic algebras and investigated in terms of
neat embedding properties. The classical results of Monk and Maddux on
non-finite axiomatizability of the classes of representable relation and
cylindric algebras of finite dimension n are reproved using also a blow up and
blur construction. Applications to n-variable fragments of first order logic
are given. The main results of the paper are summarized in tabular form at the
end of the paper.
