Around Eggleston Theorem
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916143165341696 |
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| author | Michalski, Marcin Rałowski, Robert Żeberski, Szymon |
| author_facet | Michalski, Marcin Rałowski, Robert Żeberski, Szymon |
| contents | The motivation of this work are the two classical theorems on inscribing rectangles and squares into large subsets of the plane, namely Eggleston Theorem and Mycielski Theorem.
Using Shoenfield Absoluteness Theorem we prove that for every Borel subset of the plane with uncountably many positive (with respect to measure or category) vertical section contains a rectangle $P\times B$ where $P$ is perfect and $B$ is Borel and positive. We also obtained a variant of Eggleston Theorem regarding the $σ$-ideal $\mathcal(E)$ generated by closed sets of measure zero.
Furthermore we proved that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$, where $T$ is a Spinas tree containing a Silver tree and $H$ is comeager (resp. conull). Moreover we obtained a common generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ modulo diagonal, where $T$ is a uniformly perfect tree, $H$ is comeager (resp. conull) and $[T]\subseteq H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07020 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Around Eggleston Theorem Michalski, Marcin Rałowski, Robert Żeberski, Szymon Logic Primary: 03E75, 28A05, 54H05, Secondary: 03E17 The motivation of this work are the two classical theorems on inscribing rectangles and squares into large subsets of the plane, namely Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem we prove that for every Borel subset of the plane with uncountably many positive (with respect to measure or category) vertical section contains a rectangle $P\times B$ where $P$ is perfect and $B$ is Borel and positive. We also obtained a variant of Eggleston Theorem regarding the $σ$-ideal $\mathcal(E)$ generated by closed sets of measure zero. Furthermore we proved that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$, where $T$ is a Spinas tree containing a Silver tree and $H$ is comeager (resp. conull). Moreover we obtained a common generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ modulo diagonal, where $T$ is a uniformly perfect tree, $H$ is comeager (resp. conull) and $[T]\subseteq H$. |
| title | Around Eggleston Theorem |
| topic | Logic Primary: 03E75, 28A05, 54H05, Secondary: 03E17 |
| url | https://arxiv.org/abs/2307.07020 |