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Remarks on Gitik's model and symmetric extensions on products of the Lévy collapse

We improve on results and constructions by Apter, Dimitriou, Gitik, Hayut, Karagila, and Koepke concerning large cardinals, ultrafilters, and cofinalities without the axiom of choice. In particular, we show the consistency of the following statements from certain assumptions: the first supercompact cardinal can be the first uncountable regular cardinal, all successors of regular cardinals are Ramsey, every sequence of stationary sets in ℵn is mutually stationary, an infinitary Chang conjecture holds for the cardinals ℵ2n , and all ℵn are singular. In each of the cases, our results either weaken the hypotheses or strengthen the conclusions of known proofs.