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Algebra and Discrete Mathematics, 2003, Issue 4, Pages 92–117
(Mi adm395)
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This article is cited in 1 scientific paper (total in 1 paper)
RESEARCH ARTICLE
Structural properties of extremal asymmetric colorings
Oleg Verbitsky Department of Algebra, Faculty of Mechanics and Mathematics,
Kyiv National University, Volodymyrska 60, 01033 Kyiv,
Ukraine
Abstract:
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 $\chi:\Omega\to{\{1,\dots,r\}}$ 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 $\chi^{-1}(1),\dots,\chi^{-1}(r)$ 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.
Keywords:
continuous Ramsey theory, asymmetric colorings, symmetry of a Euclidean space, polyominoes.
Citation:
Oleg Verbitsky, “Structural properties of extremal asymmetric colorings”, Algebra Discrete Math., 2003, no. 4, 92–117
Linking options:
https://www.mathnet.ru/eng/adm395 https://www.mathnet.ru/eng/adm/y2003/i4/p92
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