On Proper Colorings of Functions

TL;DR

This paper explores the properties of proper colorings of functions in infinite settings, establishing equivalences involving ultrafilters and permutations, and identifying the existence of tight colorings beyond these characterizations.

Contribution

It characterizes when a proper coloring is weakly uniform using ultrafilters and permutations, and demonstrates the existence of tight colorings not fitting this framework.

Findings

Weakly uniform colorings correspond to ultrafilters and permutations.

Existence of tight colorings that are not obtainable via ultrafilters or permutations.

Provides a characterization of proper colorings in infinite combinatorial contexts.

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.