Finite relation algebras and omitting types in modal fragments of first order logic

Let 2<n\leq l<m< \omega. Let L_n denote first order logic restricted to the first n variables. We show that the omitting types theorem fails dramatically for the n–variable fragments of first order logic with respect to clique guarded semantics, and for its packed n–variable fragments. Both are modal fragments of L_n. As a sample, we show that if there exists a finite relation algebra with a so–called strong l–blur, and no m–dimensional relational basis, then there exists a countable, atomic and complete L_n theory T and type \Gamma, such that \Gamma is realizable in every so–called m–square model of T, but any witness isolating \Gamma cannot use less than $l$ variables. An $m$–square model M of T gives a form of clique guarded semantics, where the parameter m, measures how locally well behaved M is. Every ordinary model is k–square for any n<k<\omega, but the converse is not true. Any model M is \omega–square, and the two notions are equivalent if M is countable. Such relation algebras are shown to exist for certain values of l and m like for n\leq l<\omega and m=\omega, and for l=n and m\geq n+3. The case l=n and m=\omega gives that the omitting types theorem fails for L_n with respect to (usual) Tarskian semantics: There is an atomic countable L_n theory T for which the single non–principal type consisting of co–atoms cannot be omitted in any model M of T. For n<\omega, positive results on omitting types are obained for L_n by imposing extra conditions on the theories and/or the types omitted. Positive and negative results on omitting types are obtained for infinitary variants and extensions of L_{\omega, \omega}.