论文标题
在适当的功能上
On Proper Colorings of Functions
论文作者
论文摘要
我们研究了Greenwell和Lovász的$ K $ - 交换问题的无限版本。 给定无限的红衣主教$κ$和$λ$，对于函数$ x，y \ in {}^λκ$，我们说，如果$ x（i）\ ne y（i）$（i）$ i \ inλ$完全不同。如果$ f（x）\ ne f（y）$，a函数$ f：{}^λκ\longrightArrowκ$是适当的着色，则只要$ x $和$ y $都是$ {}^λκ$的完全不同的元素。 我们说$ f $是弱统一的，如果有成对的完全不同的函数$ \ {r_α：α<κ\} \ subset {}^λκ$，使得$ f（r_α）=α$; $ f $如果没有适当的颜色$ g：{}^λκ\longrightArrowκ$，因此{}^λκ$ in {}^λκ$ a $ g（x）\ ne f（x）$恰好有一个$ x \。 我们表明，给定适当的颜色$ f：{}^λκ\至κ$，以下陈述是等效的$ f $是弱均匀的，有一个$κ^{+} $ - 完整的Ultrafiter $ \ Mathscr {U} $λ$，并且在Symmmmmmmmmmmmmmmmmmm（κ（κ）上， $ f（x）=π（α）\ \ longleftrightArrow \ \ {i \inλ：x（i）=α\} \ in \ mathscr {u} 我们还表明，有一些紧密的颜色无法获得这种方式。
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.
