A rank function for Fra\"{\i}ss\'{e} classes and the rank property

Abstract

Given a hereditary class $\mathcal{F}$ of finite relational structures, the rank function $\mathsf{rk}:\sigma\mathcal{F}\to\omega_1\cup\{\infty\}$, introduced by Kubi\'{s} and Shelah, measures how far a countable structure is from being universal within its class: $\mathsf{rk}(X)=\infty$ if and only if the Fra\"{\i}ss\'{e} 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 $\alpha\geq\omega$, then $\mathsf{rk}(\alpha)=\omega\cdot\beta_1+\lfloor\log_2 c_1\rfloor$.