Abstract: We investigate the infinite version of the k-switch problem of D. Greenwell and L. Lovász. A function F\colon ^{\lambda}\kappa \to \kappa is a proper coloring if F(x)\ne F(y) whenever x and y are totally different elements of ^{\lambda} \kappa , i.e. x(i)\ne y(i) for each i\in \lambda. We call F (i) weakly uniform if and only if there are pairwise totally different \{{r_{\alpha}:} \alpha<\kappa\}\subset ^{\lambda}\kappa with F(r_{\alpha})=\alpha; (ii) tight if no proper coloring G\colon ^{\lambda}\kappa \to \kappa differs from F at exactly one point. We prove that a proper coloring F\colon ^{\lambda}\kappa\to \kappa is weakly uniform if and only if there is a {\kappa}^{+}-complete ultrafilter {U} on \lambda and there is a permutation \pi\in {\rm Sym}(\kappa) such that for each x\in ^{\lambda}\kappa, F(x)=\pi(\alpha) \Leftrightarrow \{i\in \lambda\colon x(i)=\alpha\} \in {U}. We also show that there are tight proper colorings which cannot be obtained in this way.
Keywords: power of graph; k-switch problem; ultrafilter; tight coloring; finite independence
DOI: DOI 10.14712/1213-7243.2025.012
AMS Subject Classification: 05C76 05C63 05C15