I gave an invited talk at the conference in honor of Shelah’s 80th birthday, July 2025.
Talk Title: Marginalia to [Sh:365]
Abstract: Solovay famously proved that every stationary subset of a regular uncountable cardinal kappa may be decomposed into kappa many stationary sets. Classical variations of which are due to Ulam and Hajnal (with respect to maximally-complete ideals) and due to Shelah (with respect to club-guessing ideals). Here, we study a general case that captures and extends the above results in addition to results of Chang, Kunen-Prikry, and Eisworth. Our main result is a sought-after extension of [Sh:365, Claim 3.3] that yields an optimal extension of [Sh:365, Conclusion 4.8(2)]. Specifically, we get that at every inaccessible admitting a stationary set non-reflecting at inaccessibles, the square-bracket Ramsey relation fails with the maximal number of colors.
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.
This is joint work with Tanmay Inamdar.
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