Constrained Classes Closed Uncler Unions and ne.c. Structures

We investigate those classes \(\mathcal{K}\) of relational structures closed under operations that are defined by excluding a fixed class of finite structures. We characterize such classes and show they contain an infinite family of pairwise non-embeddable members. NEC structures are defined by certain extension conditions. We construct countable universal structures in \(\mathcal{K}\) satisfying only finitely many of the NEC extension conditions.