Structural properties of extremal asymmetric colorings

• Date: 2018
• Main Author: Verbitsky, Oleg
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2018
• Subjects: continuous Ramsey theory; asymmetric colorings; symmetry of a Euclidean space; polyominoes; 05D10
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/975
• Journal Title: Algebra and Discrete Mathematics

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,\...