Structural properties of extremal asymmetric colorings

Abstract

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 \(\chi\ : \Omega \rightarrow \{1,\ldots,r\}\) 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 \(\chi^{-1}(1),\ldots,\chi^{-1}(r)\) 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.