The externally definable Ramsey property and fixed points on type spaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912654829813760 |
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| author | Meir, Nadav Sullivan, Rob |
| author_facet | Meir, Nadav Sullivan, Rob |
| contents | We discuss the externally definable Ramsey property, a weakening of the Ramsey property for ultrahomogeneous structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for each natural number n, every subflow of the space of n-types with parameters in M has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_06522 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The externally definable Ramsey property and fixed points on type spaces Meir, Nadav Sullivan, Rob Logic Combinatorics 05C55, 03C15, 37B05, 20B27, 05D10 We discuss the externally definable Ramsey property, a weakening of the Ramsey property for ultrahomogeneous structures, where the only colourings considered are those that are externally definable: that is, definable with parameters in an elementary extension. We show a number of basic results analogous to the classical Ramsey theory, and show that, for an ultrahomogeneous structure M with countable age, the externally definable Ramsey property is equivalent to the dynamical statement that, for each natural number n, every subflow of the space of n-types with parameters in M has a fixed point. We discuss a range of examples, including results regarding the lexicographic product of structures. |
| title | The externally definable Ramsey property and fixed points on type spaces |
| topic | Logic Combinatorics 05C55, 03C15, 37B05, 20B27, 05D10 |
| url | https://arxiv.org/abs/2309.06522 |