  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.
