Labelled growth rates of $ω$-categorical structures and applications in choiceless set theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912590436761600 |
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| author | Bodor, Bertalan Braunfeld, Samuel Hanson, James E. |
| author_facet | Bodor, Bertalan Braunfeld, Samuel Hanson, James E. |
| contents | We study the labelled growth rate of an $ω$-categorical structure $\mathfrak{A}$, i.e., the number of orbits of $Aut(\mathfrak{A})$ on $n$-tuples of distinct elements, and show that the model-theoretic property of monadic stability yields a gap in the spectrum of allowable labelled growth rates. As a further application, we obtain gap in the spectrum of allowable labelled growth rates in hereditary graph classes, with no a priori assumption of $ω$-categoricity. We also establish a way to translate results about labelled growth rates of $ω$-categorical structures into combinatorial statements about sets with weak finiteness properties in the absence of the axiom of choice, and derive several results from this translation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Labelled growth rates of $ω$-categorical structures and applications in choiceless set theory Bodor, Bertalan Braunfeld, Samuel Hanson, James E. Logic Combinatorics We study the labelled growth rate of an $ω$-categorical structure $\mathfrak{A}$, i.e., the number of orbits of $Aut(\mathfrak{A})$ on $n$-tuples of distinct elements, and show that the model-theoretic property of monadic stability yields a gap in the spectrum of allowable labelled growth rates. As a further application, we obtain gap in the spectrum of allowable labelled growth rates in hereditary graph classes, with no a priori assumption of $ω$-categoricity. We also establish a way to translate results about labelled growth rates of $ω$-categorical structures into combinatorial statements about sets with weak finiteness properties in the absence of the axiom of choice, and derive several results from this translation. |
| title | Labelled growth rates of $ω$-categorical structures and applications in choiceless set theory |
| topic | Logic Combinatorics |
| url | https://arxiv.org/abs/2509.12656 |