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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| Дата: | 2003 |
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| Автор: | |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут прикладної математики і механіки НАН України
2003
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/155696 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | 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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irk-123456789-1556962019-06-18T01:28:44Z Structural properties of extremal asymmetric colorings Verbitsky, O. 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. 2003 Article 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. http://dspace.nbuv.gov.ua/handle/123456789/155696 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| 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. |
| format |
Article |
| author |
Verbitsky, O. |
| spellingShingle |
Verbitsky, O. Structural properties of extremal asymmetric colorings Algebra and Discrete Mathematics |
| author_facet |
Verbitsky, O. |
| author_sort |
Verbitsky, O. |
| 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 |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/155696 |
| 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. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT verbitskyo structuralpropertiesofextremalasymmetriccolorings |
| first_indexed |
2023-05-20T17:47:04Z |
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2023-05-20T17:47:04Z |
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1796154077442211840 |