On Proper Colorings of Functions

Abstract

We investigate the infinite version of the $k$-switch problem of Greenwell and Lov\'asz. Given infinite cardinals ${\kappa}$ and ${\lambda}$, for functions $x,y\in {}^{\lambda}\kappa $ we say that they are totally different if $x(i)\ne y(i)$ for each $i\in {\lambda}$. A function $F:{}^{\lambda}\kappa \longrightarrow {\kappa} $ is a proper coloring if $F(x)\ne F(y)$ whenever $x$ and $y$ are totally different elements of ${}^\lambda{\kappa} $. We say that $F$ is weakly uniform iff there are pairwise totally different functions $\{r_{\alpha}:{\alpha}<{\kappa}\}\subset {}^{\lambda}{\kappa}$ such that $F(r_{\alpha})={\alpha}$; $F$ is tight if there is no proper coloring $G:{}^{\lambda}\kappa \longrightarrow {\kappa}$ such that there is exactly one $x\in {}^{\lambda}{\kappa}$ with $G(x)\ne F(x)$. We show that given a proper coloring $F:{}^{\lambda}{\kappa}\to {\kappa}$, the following statements are equivalent $F$ is weakly uniform, there is a ${\kappa} ^{+}$-complete ultrafilter $\mathscr{U}$ on ${\lambda}$ and there is a permutation ${\pi}\in Symm({\kappa})$ such that for each $x\in {}^{\lambda}{\kappa}$ we have $$F(x)={\pi}({\alpha})\ \Longleftrightarrow \ \{i\in {\lambda}: x(i)={\alpha}\} \in \mathscr{U}.$$ We also show that there are tight proper colorings which cannot be obtained such a way.