A Ramsey theorem for the reals
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929362522079232 |
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| author | Inamdar, Tanmay |
| author_facet | Inamdar, Tanmay |
| contents | We prove that for every colouring of pairs of reals with finitely-many colours, there is a set homeomorphic to the rationals which takes no more than two colours. This was conjectured by Galvin in 1970, and a colouring of Sierpi{ń}ski from 1933 witnesses that the number of colours cannot be reduced to one. Previously in 1985 Shelah had shown that a stronger statement is consistent with a forcing construction assuming the existence of large cardinals. Then in 2018 Raghavan and Todorčević had proved it assuming the existence of large cardinals. We prove it in $ZFC$. In fact Raghavan and Todorčević proved, assuming more large cardinals, a similar result for a large class of topological spaces. We prove this also, again in $ZFC$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_18431 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Ramsey theorem for the reals Inamdar, Tanmay Logic Combinatorics General Topology Primary 03E02, Secondary 03E04, 03E55, 05D10, 05C55 We prove that for every colouring of pairs of reals with finitely-many colours, there is a set homeomorphic to the rationals which takes no more than two colours. This was conjectured by Galvin in 1970, and a colouring of Sierpi{ń}ski from 1933 witnesses that the number of colours cannot be reduced to one. Previously in 1985 Shelah had shown that a stronger statement is consistent with a forcing construction assuming the existence of large cardinals. Then in 2018 Raghavan and Todorčević had proved it assuming the existence of large cardinals. We prove it in $ZFC$. In fact Raghavan and Todorčević proved, assuming more large cardinals, a similar result for a large class of topological spaces. We prove this also, again in $ZFC$. |
| title | A Ramsey theorem for the reals |
| topic | Logic Combinatorics General Topology Primary 03E02, Secondary 03E04, 03E55, 05D10, 05C55 |
| url | https://arxiv.org/abs/2405.18431 |