A model for global compactness
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918251012816896 |
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| author | Jirattikansakul, Sittinon Oren, Inbar Rinot, Assaf |
| author_facet | Jirattikansakul, Sittinon Oren, Inbar Rinot, Assaf |
| contents | In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{ω+1}$ carries a uniform ultrafilter that is $θ$-indecomposable for every uncountable cardinal $θ<\aleph_ω$. In this paper, we give a global version of this result, as follows:
Assuming the consistency of a supercompact cardinal, we produce a model of set theory in which for every singular cardinal $λ$, there exists a uniform ultrafilter on $λ^+$ that is $θ$-indecomposable for every cardinal $θ$ such that $cf(λ)<θ<λ$. In our model, many instances of compactness for chromatic numbers hold, from which we infer that Hajnal's gap-1 counterexample to Hedetniemi's conjecture is best possible on the grounds of ZFC. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13584 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A model for global compactness Jirattikansakul, Sittinon Oren, Inbar Rinot, Assaf Logic Primary 03E35, Secondary 05C63, 03E55 In a classical paper by Ben-David and Magidor, a model of set theory was exhibited in which $\aleph_{ω+1}$ carries a uniform ultrafilter that is $θ$-indecomposable for every uncountable cardinal $θ<\aleph_ω$. In this paper, we give a global version of this result, as follows: Assuming the consistency of a supercompact cardinal, we produce a model of set theory in which for every singular cardinal $λ$, there exists a uniform ultrafilter on $λ^+$ that is $θ$-indecomposable for every cardinal $θ$ such that $cf(λ)<θ<λ$. In our model, many instances of compactness for chromatic numbers hold, from which we infer that Hajnal's gap-1 counterexample to Hedetniemi's conjecture is best possible on the grounds of ZFC. |
| title | A model for global compactness |
| topic | Logic Primary 03E35, Secondary 05C63, 03E55 |
| url | https://arxiv.org/abs/2412.13584 |