On Proper Colorings of Functions

Abstract

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