Universality vs Genericity and $C_4$-free graphs

TL;DR

This paper explores the relationship between universal and generic structures within classes of countable graphs, providing new examples and demonstrating that not all classes contain a generic structure, especially in the case of $C_4$-free graphs.

Contribution

It establishes that the existence of a universal structure implies a generic one for approximable classes, but not vice versa, and offers new examples of weak Fra"issé classes.

Findings

Universal structures imply generic structures in approximable classes.

The class of all countable $C_4$-free graphs lacks a generic structure.

Several new weak Fra"issé classes of finite graphs are identified.

Abstract

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.