Universality vs Genericity and C₄-free graphs

We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide several new examples of weak Fra\"iss\'e classes of finite graphs. Finally, we show that the class of all countable $C_4$-free graphs does not contain a generic structure, strengthening a result of A. Hajnal and J. Pach.