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Fix a finite ordinal n>2. We show that there exists an atomic, simple andcountable representable CA_n, such that its minimal completion is outsideSNr_nCA_{n+3}. Hence, for any finite k\geq 3, the variety SNr_nCA_{n+k} is notatom-canonical, so that the variety of CA_n's having n+k-flat representationsis not atom-canonical, too. We show, for finite k\geq 3, that S_cNr_nCA_{n+k}is not elementary, hence the class of CA_n's having complete n+3-smoothrepresentations is not elementary. We obtain analogous results by replacingflat and smooth, respectively, by (the weaker notion of) square; this give astronger result in both cases and here we can allow k to be infinite. Ourresults are proved using rainbow constructions for CA's. We lift the negativeresult on atom-canonicity to the transfinite. We also show that for any ordinal\alpha\geq \omega, for any finite k\geq 1, and for any r\in \omega, thereexists an atomic algebra A_r\in SNr_\alphaCA_{\alpha+k}\simSNr_nCA_{\alpha+k+1}, such that \Pi_{r/U} A_r\in RCA_{\alpha} where U is anynon--principal ultrafilter on \omega. Reaping the harvest of our algebraicresults we investigate a plethora of omitting types theorems for variants offirst logic including its finite variable fragments and its packed fragment.

Atom-canonicity and complete representations for cylindric-like algebras, and omitting types for the clque guarded fragment of first order logic