We assume the existence of a supercompact cardinal and produce a model with weak square but no very good scale at a particular cardinal. This follows work of Cummings, Foreman, and Magidor, but uses a different approach. We produce another model, starting from countably many supercompact cardinals, where κ,<κ holds but κ,λ fails for λ < κ.

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Aron Cummings,1,* Mohamed Osman,1,† Deepak Srivastava,2 and Madhu Menon3 1School of Electrical Engineering and Computer Science, Washington State University, Pullman, Washington 99164, USA 2NASA Ames Research Center, MST27A-1, Moffett Field, California 94035, USA 3Department of Physics and Astronomy and Center for Computational Sciences University of Kentucky, Lexington, Kentucky 40506, USA (Re...