Abstract
We prove that any suitable generalization of Laver forcing to the space κκ, for uncountable regular κ, necessarily adds a Cohen κ-real. We also study a dichotomy and an ideal naturally related to generalized Laver forcing. Using this dichotomy, we prove the following stronger result: if κ<κ = κ, then every <κ-distributive tree forcing on κκ adding a dominating κ-real which is the image of the generic under a continuous function in the ground model, adds a Cohen κ-real. This is a contribution to the study of generalized Baire spaces and answers a question from [1].
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Acknowledgments
We would like to thank Hugh Woodin and Martin Goldstern for useful discussion and advice, and the anonymous referee for his/her careful reading and helpful suggestions.
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This project was partially supported by the Isaac Newton Institute for Mathematical Sciences in the programme Mathematical, Foundational and Computational Aspects of the Higher Infinite (HIF) funded by EPSRC grant EP/K032208/1.
Supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 706219 (REGPROP).
Supported by the ÖAW Doc fellowship, and the Austrian Science Fund (FWF) under International Project number I1921.
Supported by the German Research Foundation (DFG) under Grant SP 683/4-1, and the Austrian Science Fund (FWF) under International Project numbers I4039 and Y1012.
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Khomskii, Y., Koelbing, M., Laguzzi, G. et al. Laver trees in the generalized Baire space. Isr. J. Math. 255, 599–620 (2023). https://doi.org/10.1007/s11856-022-2465-5
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DOI: https://doi.org/10.1007/s11856-022-2465-5