On indecomposable ultrafilters at every successor of singular, part 1

Indecomposability of an ultrafilter is a weakening of a completeness of an ultrafilter. Ben-David and Magidor showed that it is possible to obtain an indecomposable ultrafilter at א_{ω+1}. In our work, we extend their result to obtain indecomposable ultrafilters at every successor of a singular cardinal. This result will be the focus of our talks.
We will also provide applications in two separate directions. Firstly, we will apply one of Shelah’s results to obtain a model where the compactness of chromatic numbers of graphs holds on every successor of a singular. We will also apply a proof from Raghavan and Shelah to show that for any fixed successor of a singular, it is consistent to have a small ultrafilter number at that cardinal. Secondly, we will give a few examples where Silver type theorems fail.

This is joint work with Sittinon Jirattikansakul and Assaf Rinot