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- W4309797003 abstract "We investigate the infinite version of $k$-switch problem of Greenwell and Lov'asz. Given infinite cardinals ${kappa}$ and ${lambda}$, for functions $x,yin {}^{kappa}lambda $ we say that they are totally different if $x(i)ne y(i)$ for each $iin {lambda}$. 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 (i) $F$ is weakly uniform iff there are pairwise totally different functions ${r_{alpha}:{alpha}<{kappa}}subset {}^{lambda}{kappa}$ such that $F(r_{alpha})={alpha}$; (ii) $F$ is tight if there is no proper coloring $G:{}^{lambda}kappa longrightarrow {kappa}$ such that there is exactly one $xin {}^{lambda}{kappa}$ with $G(x)ne F(x)$. We show that given a proper coloring $F:{}^{lambda}{kappa}to {kappa}$, the following statements are equivalent (i) $F$ is weakly uniform, (ii) there is a ${kappa}$ complete ultrafilter $mathscr{U}$ on ${lambda}$ an there is a permutation ${pi}in Symm({kappa})$ such that for each $xin ^{lambda}{kappa}$ we have $$F(x)={pi}({alpha}) Longleftrightarrow {iin {lambda}: x(i)={alpha}} in mathscr{U}.$$ We also show that there are tight proper colorings which can not be obtain such a way." @default.
- W4309797003 created "2022-11-29" @default.
- W4309797003 creator A5068767867 @default.
- W4309797003 date "2022-11-19" @default.
- W4309797003 modified "2023-09-23" @default.
- W4309797003 title "On Proper Colorings of Functions" @default.
- W4309797003 doi "https://doi.org/10.48550/arxiv.2211.10654" @default.
- W4309797003 hasPublicationYear "2022" @default.
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