WAS ULAM RIGHT? III: INDECOMPOSABLE IDEALS

Abstract We continue our study of Ulam’s measure problem. In contrast to our previous works, we shift our focus from measures stratified by their additivity, to measures stratified by their indecomposability. The breakthrough here is obtained by replacing the classical ‘least’ function associated with ideals by a two-dimensional ‘last’ function associated with walks on ordinals. Consequently, we obtain conditions under which a measure admits not just infinite pairwise disjoint families of positive sets, but in fact families of maximum possible size. As an application we solve a problem left open in Shelah’s Cardinal Arithmetic book, proving that for every weakly inaccessible cardinal $\kappa $ , if there exists a stationary subset of $\kappa $ that does not reflect at regulars, then the strong Ramsey relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ holds.