论文标题
乌拉姆对吗？ II：宽度和一般理想
Was Ulam right? II: Small width and general ideals
论文作者
论文摘要
我们继续研究Sierpinski型着色。与前传论文相反，我们在这里专注于以其完整性分层的理想色彩。特别是，改善了乌拉姆（Ulam）的定理及其扩展为hajnal，可以证明，如果$κ$是常规的无数基数，在L中并不弱紧凑，那么有一个普遍的见证人，即$κ$ complete的理想是非饱和的。具体而言，$κ$ - 许多分解为$κ$，因此，对于每一个$κ$ complete of $κ$上的理想$ j $，而每$ b \ in J^+$中的每一个$ b \ n in j^+$，这是$κ$ b $ to $κ$ - many $ j^+$ sets的$ b $ b $之一。 这里的第二个重点是色彩狭窄的特征，它已经存在于Sierpinski定理中。此功能可确保适合理想的着色也适用于所有具有必要完整性的上级。事实证明，与常客的继任者不同，每个奇异的枢机主教的每个后继者都承认着色如此狭窄。
We continue our study of Sierpinski-type colourings. In contrast to the prequel paper, we focus here on colourings for ideals stratified by their completeness degree. In particular, improving upon Ulam's theorem and its extension by Hajnal, it is proved that if $κ$ is a regular uncountable cardinal that is not weakly compact in L, then there is a universal witness for non-weak-saturation of $κ$-complete ideals. Specifically, there are $κ$-many decompositions of $κ$ such that, for every $κ$-complete ideal $J$ over $κ$, and every $B\in J^+$, one of the decompositions shatters $B$ into $κ$-many $J^+$-sets. A second focus here is the feature of narrowness of colourings, one already present in the theorem of Sierpinski. This feature ensures that a colouring suitable for an ideal is also suitable for all superideals possessing the requisite completeness degree. It is proved that unlike successors of regulars, every successor of a singular cardinal admits such a narrow colouring.
