Towards a Ryll-Nardzewski-type Theorem for weakly oligomorphic structures

A structure is called weakly oligomorphic if it realizes only finitely many n-ary positive existential types for every n. The goal of this paper is to show that the notions of homomorphism-homogeneity, and weak oligomorphy are not only completely analogous to the classical notions of ultrahomogeneity and oligomorphy, but are actually closely related. A first result is a Fra\"iss\'e-type theorem for homomorphism-homogeneous relational structures. Further we show that every weakly oligomorphic homomorphism-homogeneous structure contains (up to isomorphism) a unique homogeneous, homomorphism-homogeneous core, to which it is homomorphism-equivalent. As a consequence, we obtain that every countable weakly oligomorphic structure is homomorphism-equivalent with a finite or \omega-categorical structure. Another result is the characterization of positive existential theories of weakly oligomorphic structures as the positive existential parts of \omega-categorical theories. Finally, we show, that the countable models of countable weakly oligomorphic structures are mutually homomorphism-equivalent (we call first order theories with this property weakly \omega-categorical). These results are in analogy with part of the Engeler-Ryll-Nardzewski-Svenonius-theorem.