On Proper Colorings of Functions

TL;DR

This work generalizes the finite Greenwell–Lovász k-switch theorem to the infinite setting by studying proper colorings F: $^{\lambda}\kappa\to\mu$. It identifies key notions—weak uniformity and tightness—and proves that a weakly uniform coloring corresponds to a $\kappa^{+}$-complete ultrafilter on $\lambda$ together with a permutation on $\kappa$, via $F(x)=\pi(\alpha)$ whenever $\{i: x(i)=\alpha\}\in\mathscr{U}$. It further shows that tight colorings exist beyond this representation and provides a detailed classification of $2$-tight colorings through ultrafilter quotients, along with a study of finitely independent colorings and forcing constructions under CH. The paper also supplies constructive methods, including color partitions and maximal lawful sets, to produce nontrivial tight colorings that are not captured by the ultrafilter-based form, thereby enriching the landscape of infinite switch-type colorings and their combinatorial structure.

Abstract

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

Theorems & Definitions (103)

• Definition 1.1
• Theorem 1.2: KoTo
• proof
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Definition 3.4
• Theorem 3.5