Asymptotic Theories of Classes Defined by Forbidden Homomorphisms
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866916295973273600 |
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| author | Bodirsky, Manuel Jahel, Colin |
| author_facet | Bodirsky, Manuel Jahel, Colin |
| contents | We study the first-order almost-sure theories for classes of finite structures that are specified by homomorphically forbidding a set $\mathcal{F}$ of finite structures. If $\mathcal{F}$ consists of undirected graphs, a full description of these theories can be derived from the Kolaitis-Prömel-Rothschild theorem, which treats the special case where $\mathcal{F} = \{K_n\}$. The corresponding question for finite sets $\mathcal{F}$ of finite directed graphs is wide open. We present a full description of the almost-sure theories of classes described by homomorphically forbidding finite sets $\mathcal{F}$ of oriented trees; all of them are $ω$-categorical. In our proof, we establish a result of independent interest, namely that every constraint satisfaction problem for a finite digraph has first-order convergence, and that the corresponding asymptotic theory can be described as a finite linear combination of $ω$-categorical theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_01404 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Asymptotic Theories of Classes Defined by Forbidden Homomorphisms Bodirsky, Manuel Jahel, Colin Combinatorics Logic We study the first-order almost-sure theories for classes of finite structures that are specified by homomorphically forbidding a set $\mathcal{F}$ of finite structures. If $\mathcal{F}$ consists of undirected graphs, a full description of these theories can be derived from the Kolaitis-Prömel-Rothschild theorem, which treats the special case where $\mathcal{F} = \{K_n\}$. The corresponding question for finite sets $\mathcal{F}$ of finite directed graphs is wide open. We present a full description of the almost-sure theories of classes described by homomorphically forbidding finite sets $\mathcal{F}$ of oriented trees; all of them are $ω$-categorical. In our proof, we establish a result of independent interest, namely that every constraint satisfaction problem for a finite digraph has first-order convergence, and that the corresponding asymptotic theory can be described as a finite linear combination of $ω$-categorical theories. |
| title | Asymptotic Theories of Classes Defined by Forbidden Homomorphisms |
| topic | Combinatorics Logic |
| url | https://arxiv.org/abs/2204.01404 |