Structural properties of extremal asymmetric colorings

Let $\Omega$ be a space with probability measure $\mu$ for which the notion of symmetry is defined. Given $A\subseteq\Omega$, let $ms(A)$ denote the supremum of $\mu(B)$ over symmetric $B\subseteq A$. An $r$-coloring of $\Omega$ is a measurable map $\chi:\Omega\to{\{1,\dots,r\}}$ possibly undefined on a set of measure 0. Given an $r$-coloring $\chi$, let $ms(\Omega;\chi)=\max_{1\le i\le r}ms(\chi^{-1}(i))$. With each space $\Omega$ we associate a Ramsey type number $ms(\Omega,r)=\inf_\chi ms(\Omega;\chi)$. We call a coloring $\chi$ congruent if the monochromatic classes $\chi^{-1}(1),\dots,\chi^{-1}(r)$ are pairwise congruent, i.e., can be mapped onto each other by a symmetry of $\Omega$. We define $ms^{\star}(\Omega,r)$ to be the infimum of $ms(\Omega;\chi)$ over congruent $\chi$. We prove that $ms(S^1,r)=ms^{\star}(S^1,r)$ for the unitary circle $S^1$ endowed with standard symmetries of a plane, estimate $ms^{\star}([0,1),r)$ for the unitary interval of reals considered with central symmetry, and explore some other regularity properties of extremal colorings for various spaces.