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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| Datum: | 2018 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Lugansk National Taras Shevchenko University
2018
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| Schlagworte: | |
| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/975 |
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| Назва журналу: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Zusammenfassung: | 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. |
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