Was Ulam right? II: small width and general ideals

We continue our study of Sierpiński-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 Sierpiński. 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.