Homomorphism and Embedding Universal Structures for Restricted Classes

This paper unifies problems and results related to (embedding) universal and homomorphism universal structures. On the one side we give a new combinatorial proof of the existence of universal objects for homomorphism defined classes of structures (thus reproving a result of [6]) and on the other side this leads to the new proof of the existence of dual objects (established by [32]). Our explicite approach has further applications to special structures such as variants of the rational Urysohn space. We also solve a related extremal problem which shows the optimality (of the used lifted arities) of our construction. Our method also relates to weakly indivisible homomorphism defined classes of structures.