A rank function for Fraïssé classes and the rank property

Given a hereditary class $\mathcal{F}$ of finite relational structures, the rank function $\mathsf{r}\mathsf{k}:\sigma \mathcal{F}\to {\omega }_1\cup \{\mathrm{\infty }\}$, introduced by Kubiś and Shelah, measures how far a countable structure is from being universal within its class: $\mathsf{r}\mathsf{k}(X)=\mathrm{\infty }$ if and only if the Fraïssé limit embeds into $X$. We say that $\mathcal{F}$ has the Rank Property (RP) if every countable ordinal is realized as the rank of some $X\in \sigma \mathcal{F}$. We develop the basic theory of the rank function and establish RP for three families of classes: those satisfying the free amalgamation property and the full extension property (covering graphs, hypergraphs, and many others); finite tournaments; and finite linear orders. For the latter, we compute the rank of every countable ordinal: if ${\omega }^{{\beta }_1}\cdot c_1$ is the leading Cantor normal form term of $\alpha \ge \omega$, then $\mathsf{r}\mathsf{k}(\alpha )=\omega \cdot {\beta }_1+\lfloor {\log }_2{c}_1\rfloor$.