Structural properties of extremal asymmetric colorings
Let \(\Omega\) be a space with probability measure \(\mu\) for which the notion of symmetry is defined. Given \(A\subseteq\Omega\), let \( ms(A)\) denote the supremum of \(\mu(B)\) over symmetric \(B\subseteq A\). An \(r\)-coloring of \(\Omega\) is a measurable map \(\chi\ : \Omega \rightarrow \{1,\...
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| Дата: | 2018 |
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| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2018
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/975 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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admjournalluguniveduua-article-9752018-05-14T07:18:11Z Structural properties of extremal asymmetric colorings Verbitsky, Oleg continuous Ramsey theory, asymmetric colorings, symmetry of a Euclidean space, polyominoes 05D10 Let \(\Omega\) be a space with probability measure \(\mu\) for which the notion of symmetry is defined. Given \(A\subseteq\Omega\), let \( ms(A)\) denote the supremum of \(\mu(B)\) over symmetric \(B\subseteq A\). An \(r\)-coloring of \(\Omega\) is a measurable map \(\chi\ : \Omega \rightarrow \{1,\ldots,r\}\) possibly undefined on a set of measure 0. Given an \(r\)-coloring \(\chi\), let \(ms(\Omega;\chi)=\max_{1\le i\le r} ms(\chi^{-1}(i))\). With each space \(\Omega\) we associate a Ramsey type number \(ms(\Omega,r)=\inf_\chi ms(\Omega;\chi)\). We call a coloring \(\chi\) congruent if the monochromatic classes \(\chi^{-1}(1),\ldots,\chi^{-1}(r)\) are pairwise congruent, i.e., can be mapped onto each other by a symmetry of \(\Omega\). We define \( ms^\star(\Omega,r)\) to be the infimum of \(ms(\Omega;\chi)\) over congruent \(\chi\).We prove that \( ms(S^1,r)=ms^\star(S^1,r)\) for the unitary circle \(S^1\) endowed with standard symmetries of a plane, estimate \(ms^\star([0,1),r)\) for the unitary interval of reals considered with central symmetry, and explore some other regularity properties of extremal colorings for various spaces. Lugansk National Taras Shevchenko University 2018-05-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/975 Algebra and Discrete Mathematics; Vol 2, No 4 (2003) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/975/504 Copyright (c) 2018 Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics |
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| datestamp_date |
2018-05-14T07:18:11Z |
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OJS |
| language |
English |
| topic |
continuous Ramsey theory asymmetric colorings symmetry of a Euclidean space polyominoes 05D10 |
| spellingShingle |
continuous Ramsey theory asymmetric colorings symmetry of a Euclidean space polyominoes 05D10 Verbitsky, Oleg Structural properties of extremal asymmetric colorings |
| topic_facet |
continuous Ramsey theory asymmetric colorings symmetry of a Euclidean space polyominoes 05D10 |
| format |
Article |
| author |
Verbitsky, Oleg |
| author_facet |
Verbitsky, Oleg |
| author_sort |
Verbitsky, Oleg |
| title |
Structural properties of extremal asymmetric colorings |
| title_short |
Structural properties of extremal asymmetric colorings |
| title_full |
Structural properties of extremal asymmetric colorings |
| title_fullStr |
Structural properties of extremal asymmetric colorings |
| title_full_unstemmed |
Structural properties of extremal asymmetric colorings |
| title_sort |
structural properties of extremal asymmetric colorings |
| description |
Let \(\Omega\) be a space with probability measure \(\mu\) for which the notion of symmetry is defined. Given \(A\subseteq\Omega\), let \( ms(A)\) denote the supremum of \(\mu(B)\) over symmetric \(B\subseteq A\). An \(r\)-coloring of \(\Omega\) is a measurable map \(\chi\ : \Omega \rightarrow \{1,\ldots,r\}\) possibly undefined on a set of measure 0. Given an \(r\)-coloring \(\chi\), let \(ms(\Omega;\chi)=\max_{1\le i\le r} ms(\chi^{-1}(i))\). With each space \(\Omega\) we associate a Ramsey type number \(ms(\Omega,r)=\inf_\chi ms(\Omega;\chi)\). We call a coloring \(\chi\) congruent if the monochromatic classes \(\chi^{-1}(1),\ldots,\chi^{-1}(r)\) are pairwise congruent, i.e., can be mapped onto each other by a symmetry of \(\Omega\). We define \( ms^\star(\Omega,r)\) to be the infimum of \(ms(\Omega;\chi)\) over congruent \(\chi\).We prove that \( ms(S^1,r)=ms^\star(S^1,r)\) for the unitary circle \(S^1\) endowed with standard symmetries of a plane, estimate \(ms^\star([0,1),r)\) for the unitary interval of reals considered with central symmetry, and explore some other regularity properties of extremal colorings for various spaces. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2018 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/975 |
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AT verbitskyoleg structuralpropertiesofextremalasymmetriccolorings |
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2025-12-02T15:43:46Z |
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2025-12-02T15:43:46Z |
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