  Let n be finite >2. We show that any class between S\Nr_n\CA_{n+3} and RCA_n
is not atom canonical, and any class containing the class of completely
representable algebras and contained in S_c\Nr_n\CA_{n+3} is not elementary. We
show that there is no finite variable universal axiomatization of many diagonal
free reducts of representable cylindric algebras of dimension n, like the
varieties of representable diagonal-free cylindric algebras and Halmos'
polyadic algebras (without equality). We apply our hitherto obtained algebraic
results to show that the omitting types theorem fails for finite variable
fragments of first order logic with and without equality, having n variables,
even if we count in severely relativized models as candidates for omitting
single non-principle types. Finally, we show that for many cylindric-like
algebras, like diagonal free cylindric algebras and Halmos' polyadic algebras
with and without equality the class of strongly representable atom structures
of finite dimension >2 is not elementary.
