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.