乌拉姆是对的吗？II：小宽度与一般理想
Was Ulam right? II: Small width and general ideals
AI总结：
本文研究按完全度分层的理想的谢尔宾斯基型染色，改进了Ulam定理及Hajnal的推广，证明了L中非弱紧的正则不可数基数上存在κ-完全理想非弱饱和的通用见证，且奇异基数的后继容许窄染色。
摘要：
我们继续研究谢尔宾斯基（Sierpinski）型染色。与前篇论文不同，本文聚焦于按完全度分层的理想的染色。具体而言，我们改进了乌拉姆（Ulam）定理及其由豪伊瑙尔（Hajnal）推广的结果，证明了：若$κ$是一个正则不可数基数，且在可构造宇宙L中不是弱紧的，则存在$κ$-完全理想非弱饱和的通用见证。更确切地说，存在$κ$种对$κ$的分解，使得对于每个$κ$上的$κ$-完全理想$J$，以及每个$B∈J^+$，其中一种分解可将$B$拆分为$κ$个$J^+$-集。\n本文的第二个研究重点是染色的窄性特征，这一特征早已出现在谢尔宾斯基的定理中。该特征确保了适用于某个理想的染色，同样适用于所有具备所需完全度的超理想。我们证明了，与正则基数的后继不同，每个奇异基数的后继都容许这样的窄染色。
英文摘要：
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.
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