Construction of models of higher Chang conjectures and saturated ideals

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1. On the construction of a model in which higher Chang conjecture holds. Foreman constructed a model in which Chang conjecture holds for triples, but the case for quadruples remains open. As the first step toward the solution, we gave a much simpler construction of a model like Foreman's.2. On the construction of a model that contains a saturated ideal. Woodin constructed a model that contains a dense ideal, but the corresponding Boolean algebra is not yet known. As the first step toward the solution, we constructed by two-step iterated forcing, a model that contains a strongly saturated ideal originally due to Laver, and that contains a centered ideal originally due to Laver and Foreman.