Structural properties of extremal asymmetric colorings

Please use this identifier to cite or link to this item: http://hdl.handle.net/123456789/81

Title:	Structural properties of extremal asymmetric colorings
Authors:	Verbitsky, Oleg
Issue Date:	2003
Publisher:	Луганский национальный университет им. Т. Шевченко
Description:	Let be a space with probability measure μ for which the notion of symmetry is defined. Given A µ , let ms(A) denote the supremum of μ(B) over symmetric B µ A. An r-coloring of is a measurable map Â : ! {1, . . . , r} possi- bly undefined on a set of measure 0. Given an r-coloring Â, let ms(; Â) = max1·i·r ms(Â−1(i)). With each space we associate a Ramsey type number ms(, r) = infÂms(; Â). We call a col- oring Â congruent if the monochromatic classes Â−1(1), . . . , Â−1(r) are pairwise congruent, i.e., can be mapped onto each other by a symmetry of . We define ms?(, r) to be the infimum of ms(; Â) over congruent Â. We prove that ms(S1, r) = ms?(S1, r) for the unitary circle S1 endowed with standard symmetries of a plane, estimate 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.
URI:	http://hdl.handle.net/123456789/81
Appears in Collections:	Статті

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