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Structural properties of extremal asymmetric colorings

Show simple item record Verbitsky, Oleg 2015-10-21T10:49:30Z 2015-10-21T10:49:30Z 2003
dc.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  : ­ ! {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  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. uk_UA
dc.language.iso en uk_UA
dc.publisher Луганский национальный университет им. Т. Шевченко uk_UA
dc.title Structural properties of extremal asymmetric colorings uk_UA
dc.type Article uk_UA

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    This is a special issue of the journal devoted to the jubilee of the outstanding Ukrainian and Russian mathematician, one of the founder of our journal Professor Rostislav Grigorchuk, who was 50 years of age on February 23, 2003.

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