论文标题
分区上的颜色很强
Strong colorings over partitions
论文作者
论文摘要
红衣主教$κ$上的强烈着色是$ f：[κ]^2 \至κ$，使得每$ a \subseteqκ$ of Gull Size $κ$，每种颜色$γ<κ$都由$ f \ upharpoonright [a]^2 $实现。符号$κ\ nrightArrow [κ]^2_κ$断言$κ$上的颜色很强。 我们介绍了符号$κ\ nrightArrow_p [κ]^2_κ$，该符号宣称存在着色$ f：[κ]^2 \ toκ$，这在分区$ p：[κ]^2 \toθ$上很强。如果对于[κ]^κ$中的每一个$ a \，$ f $都有$ i <θ$的颜色强，以便每种颜色$γ<κ$都可以由$ f \ upharpoonright（[a]^2 \ cap p^{ - 1}（i））$。 我们证明，每当$κ\ nrightArrow [κ]^2_κ$所容纳时，$κ\ nrightArrow_p [κ]^2_κ$都会适用于任意有限分区$ p $。同样，任意有限的$ p $ -s可以添加到任何ZFC中包含的更强符号中。如果$κ^θ=κ$，则$κ\ nrightArrow_p [κ]^2_κ$和更牢固的符号，例如$ \ mathrm {pr} _1（Pr} _1（κ，κ，κ，κ，κ，χ）$或$ \ Mathrm {pr} 部分。
A strong coloring on a cardinal $κ$ is a function $f:[κ]^2\to κ$ such that for every $A\subseteq κ$ of full size $κ$, every color $γ<κ$ is attained by $f\upharpoonright[A]^2$. The symbol $κ\nrightarrow [κ]^2_κ$ asserts the existence of a strong coloring on $κ$. We introduce the symbol $κ\nrightarrow_p[κ]^2_κ$ which asserts the existence of a coloring $f:[κ]^2\to κ$ which is strong over a partition $p:[κ]^2\toθ$. A coloring $f$ is strong over $p$ if for every $A\in [κ]^κ$ there is $i<θ$ so that every color $γ<κ$ is attained by $f\upharpoonright ([A]^2\cap p^{-1}(i))$. We prove that whenever $κ\nrightarrow[κ]^2_κ$ holds, also $κ\nrightarrow_p[κ]^2_κ$ holds for an arbitrary finite partition $p$. Similarly, arbitrary finite $p$-s can be added to stronger symbols which hold in any model of ZFC. If $κ^θ=κ$, then $κ\nrightarrow_p[κ]^2_κ$ and stronger symbols, like $\mathrm{Pr}_1(κ,κ,κ,χ)$ or $\mathrm{Pr}_0(κ,κ,κ,\aleph_0)$, hold also for an arbitrary partition $p$ to $θ$ parts.
