Reconstruction of a Coloring from its Homogeneous Sets

Abstract

AbstractWe study the following reconstruction problem for colorings. Given a countable set X (finite or infinite), a coloring on X is a function $$\varphi : [X]^{2}\rightarrow \{0,1\}$$ φ : [ X ] 2 → { 0 , 1 } , where $$[X]^{2}$$ [ X ] 2 is the collection of all 2-elements subsets of X. A set $$H\subseteq X$$ H ⊆ X is homogeneous for $$\varphi$$ φ when $$\varphi$$ φ is constant on $$[H]^2$$ [ H ] 2 . Let $${{\,\textrm{hom}\,}}(\varphi )$$ hom ( φ ) be the collection of all homogeneous sets for $$\varphi$$ φ . The coloring $$1-\varphi$$ 1 - φ is called the complement of $$\varphi$$ φ . We say that $$\varphi$$ φ is reconstructible up to complementation from its homogeneous sets, if for any coloring $$\psi$$ ψ on X such that $${{\,\textrm{hom}\,}}(\varphi )={{\,\textrm{hom}\,}}(\psi )$$ hom ( φ ) = hom ( ψ ) we have that either $$\psi =\varphi$$ ψ = φ or $$\psi =1-\varphi$$ ψ = 1 - φ . We present several conditions for reconstructibility and non reconstructibility. For X an infinite countable set, we show that there is a Borel way to recovering a coloring from its homogeneous sets.