Structural properties of extremal asymmetric colorings
Let Ω be a space with probability measure µ for which the notion of symmetry is defined. Given A ⊆ Ω, let ms(A) denote the supremum of µ(B) over symmetric B ⊆ A. An r-coloring of Ω is a measurable map χ : Ω → {1, . . . , r} possibly undefined on a set of measure 0. Given an r-coloring χ, let ms(Ω; χ...
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| Published in: | Algebra and Discrete Mathematics |
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| Date: | 2003 |
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| Format: | Article |
| Language: | English |
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Інститут прикладної математики і механіки НАН України
2003
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/155696 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Structural properties of extremal asymmetric colorings / O. Verbitsky // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 92–117. — Бібліогр.: 12 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862734876553248768 |
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| author | Verbitsky, O. |
| author_facet | Verbitsky, O. |
| citation_txt | Structural properties of extremal asymmetric colorings / O. Verbitsky // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 92–117. — Бібліогр.: 12 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | Let Ω be a space with probability measure µ for which the notion of symmetry is defined. Given A ⊆ Ω, let ms(A) denote the supremum of µ(B) over symmetric B ⊆ A. An r-coloring of Ω is a measurable map χ : Ω → {1, . . . , r} possibly undefined on a set of measure 0. Given an r-coloring χ, let ms(Ω; χ) = max₁≤i≤r ms(χ⁻¹ (i)). With each space Ω we associate a Ramsey type number ms(Ω, r) = infχ ms(Ω; χ). We call a coloring χ congruent if the monochromatic classes χ⁻¹ (1), . . . , χ⁻¹ (r) are pairwise congruent, i.e., can be mapped onto each other by a symmetry of Ω. We define ms* (Ω, r) to be the infimum of ms(Ω; χ) over congruent χ. We prove that ms(S¹ , r) = ms* ([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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| first_indexed | 2025-12-07T19:45:28Z |
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| id | nasplib_isofts_kiev_ua-123456789-155696 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T19:45:28Z |
| publishDate | 2003 |
| publisher | Інститут прикладної математики і механіки НАН України |
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| spelling | Verbitsky, O. 2019-06-17T10:52:28Z 2019-06-17T10:52:28Z 2003 Structural properties of extremal asymmetric colorings / O. Verbitsky // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 4. — С. 92–117. — Бібліогр.: 12 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 05D10. https://nasplib.isofts.kiev.ua/handle/123456789/155696 Let Ω be a space with probability measure µ for which the notion of symmetry is defined. Given A ⊆ Ω, let ms(A) denote the supremum of µ(B) over symmetric B ⊆ A. An r-coloring of Ω is a measurable map χ : Ω → {1, . . . , r} possibly undefined on a set of measure 0. Given an r-coloring χ, let ms(Ω; χ) = max₁≤i≤r ms(χ⁻¹ (i)). With each space Ω we associate a Ramsey type number ms(Ω, r) = infχ ms(Ω; χ). We call a coloring χ congruent if the monochromatic classes χ⁻¹ (1), . . . , χ⁻¹ (r) are pairwise congruent, i.e., can be mapped onto each other by a symmetry of Ω. We define ms* (Ω, r) to be the infimum of ms(Ω; χ) over congruent χ. We prove that ms(S¹ , r) = ms* ([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. I am thankful to Yaroslav Vorobets whose insightful suggestions contributed a lot to this work. I would like also to thank Taras Banakh
 and Alexander Ravsky for helpful discussions and useful pointers to the
 literature. I especially thank Alexander Ravsky for careful proofreading
 of the manuscript and allowing me to announce here his Theorem 5.13. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Structural properties of extremal asymmetric colorings Article published earlier |
| spellingShingle | Structural properties of extremal asymmetric colorings Verbitsky, O. |
| title | 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_short | Structural properties of extremal asymmetric colorings |
| title_sort | structural properties of extremal asymmetric colorings |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/155696 |
| work_keys_str_mv | AT verbitskyo structuralpropertiesofextremalasymmetriccolorings |