Was Ulam right? II: Small width and general ideals

TL;DR

This paper advances the understanding of colourings related to ideals, especially for regular uncountable cardinals, by extending Ulam's theorem and exploring narrow colourings for successors of singular cardinals.

Contribution

It proves the existence of universal witnesses for non-weak-saturation of certain ideals and establishes the narrowness property for colourings of successors of singular cardinals.

Findings

Existence of $oldsymbol{ ext{κ}}$-many decompositions shattering any positive set.

Universal witness for non-weak-saturation of $ ext{κ}$-complete ideals.

Every successor of a singular cardinal admits a narrow colouring.

Abstract

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 $\kappa$ is a regular uncountable cardinal that is not weakly compact in L, then there is a universal witness for non-weak-saturation of $\kappa$-complete ideals. Specifically, there are $\kappa$-many decompositions of $\kappa$ such that, for every $\kappa$-complete ideal $J$ over $\kappa$, and every $B\in J^+$, one of the decompositions shatters $B$ into $\kappa$-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.