Structural properties of extremal asymmetric colorings

Abstract

Let Ω be a space with probability measure µ for which the notion of symmetry is defined. Given A ⊆ Ω, let ms(A) denote the supremum of µ(B) over symmetric B ⊆ A. An r-coloring of Ω is a measurable map χ : Ω → {1, . . . , r} possibly undefined on a set of measure 0. Given an r-coloring χ, let ms(Ω; χ) = max₁≤i≤r ms(χ⁻¹ (i)). With each space Ω we associate a Ramsey type number ms(Ω, r) = infχ ms(Ω; χ). We call a coloring χ congruent if the monochromatic classes χ⁻¹ (1), . . . , χ⁻¹ (r) are pairwise congruent, i.e., can be mapped onto each other by a symmetry of Ω. We define ms* (Ω, r) to be the infimum of ms(Ω; χ) over congruent χ. We prove that ms(S¹ , r) = 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.