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

TL;DR

This paper introduces a rank function for hereditary classes of finite relational structures, analyzing its properties and establishing the Rank Property for various classes, including graphs, hypergraphs, tournaments, and linear orders.

Contribution

It develops the theory of the rank function and proves the Rank Property for classes with free amalgamation, extension properties, tournaments, and linear orders, including explicit rank calculations.

Findings

Established the Rank Property for classes with free amalgamation and extension properties.

Computed the rank of every countable ordinal for finite linear orders.

Developed the basic theory of the rank function for hereditary classes.

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 $\mathsf{rk}(\alpha)=\omega\cdot\beta_1+\lfloor\log_2 c_1\rfloor$.