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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.14461 |
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| _version_ | 1866909034821451776 |
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| author | López-Callejas, Carlos Navarro-Castillo, Jareb |
| author_facet | López-Callejas, Carlos Navarro-Castillo, Jareb |
| contents | Given a hereditary class $\mathcal{F}$ of finite relational structures, the rank function $\mathsf{rk}:σ\mathcal{F}\toω_1\cup\{\infty\}$, introduced by Kubiś 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\"ıssé 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 $X\inσ\mathcal{F}$.
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 $ω^{β_1}\cdot c_1$ is the leading Cantor normal form term of $α\geqω$, then $\mathsf{rk}(α)=ω\cdotβ_1+\lfloor\log_2 c_1\rfloor$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A rank function for Fra\"ıssé classes and the rank property López-Callejas, Carlos Navarro-Castillo, Jareb Logic Given a hereditary class $\mathcal{F}$ of finite relational structures, the rank function $\mathsf{rk}:σ\mathcal{F}\toω_1\cup\{\infty\}$, introduced by Kubiś 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\"ıssé 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 $X\inσ\mathcal{F}$. 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 $ω^{β_1}\cdot c_1$ is the leading Cantor normal form term of $α\geqω$, then $\mathsf{rk}(α)=ω\cdotβ_1+\lfloor\log_2 c_1\rfloor$. |
| title | A rank function for Fra\"ıssé classes and the rank property |
| topic | Logic |
| url | https://arxiv.org/abs/2604.14461 |