Year: 2026
Type: article
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 $\kappa $ κ , if there exists a stationary subset of kappa $\kappa $ κ that does not reflect at regulars, then the strong Ramsey relation kappa right arrow with stroke kappa Subscript kappa Superscript 2 $\kappa \nrightarrow [\kappa ]^2_\kappa $ κ ↛ κ κ 2 holds.
Authors: Tanmay Inamdar, Assaf Rinot
Institutions: Ben-Gurion University of the Negev, Bar-Ilan University
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
Funder: Israel Science Foundation
Award: G6663309099
OpenAlex ID: W7154993481