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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A framework for forcing constructions at successors of singular cardinals
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by James Cummings, Mirna Džamonja, Menachem Magidor, Charles Morgan and Saharon Shelah PDF
Trans. Amer. Math. Soc. 369 (2017), 7405-7441 Request permission

Abstract:

We describe a framework for proving consistency results about singular cardinals of arbitrary cofinality and their successors. This framework allows the construction of models in which the Singular Cardinals Hypothesis fails at a singular cardinal $\kappa$ of uncountable cofinality, while $\kappa ^+$ enjoys various combinatorial properties.

As a sample application, we prove the consistency (relative to that of ZFC plus a supercompact cardinal) of there being a strong limit singular cardinal $\kappa$ of uncountable cofinality where SCH fails and such that there is a collection of size less than $2^{\kappa ^+}$ of graphs on $\kappa ^+$ such that any graph on $\kappa ^+$ embeds into one of the graphs in the collection.

References
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Additional Information
  • James Cummings
  • Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
  • MR Author ID: 289375
  • ORCID: 0000-0002-7913-0427
  • Email: jcumming@andrew.cmu.edu
  • Mirna Džamonja
  • Affiliation: School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom
  • ORCID: setImmediate$0.3709267400444315$1
  • Email: M.Dzamonja@uea.ac.uk
  • Menachem Magidor
  • Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Givat Ram, Israel
  • MR Author ID: 118010
  • ORCID: 0000-0002-5568-8397
  • Email: mensara@savion.huji.ac.il
  • Charles Morgan
  • Affiliation: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom — and — School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, United Kingdom
  • MR Author ID: 290043
  • Email: charles.morgan@ucl.ac.uk
  • Saharon Shelah
  • Affiliation: Institute of Mathematics, Hebrew University of Jerusalem, 91904 Givat Ram, Israel — and — Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email: shelah@math.huji.ac.il
  • Received by editor(s): March 25, 2014
  • Received by editor(s) in revised form: October 9, 2015, April 1, 2016, and April 30, 2016
  • Published electronically: May 31, 2017
  • Additional Notes: The first author thanks the National Science Foundation for support through grant DMS-1101156. The first, second and fourth authors thank the Institut Henri Poincaré for support through the “Research in Paris” program during the period June 24–29, 2013. The second author thanks EPSRC for support through grants EP/G068720 and EP/I00498. The second, third and fifth authors thank the Mittag-Leffler Institute for support during the month of September 2009. The fourth author thanks EPSRC for support through grant EP/I00498. This publication is denoted [Sh963] in Saharon Shelah’s list of publications. Shelah thanks the United States-Israel Binational Science Foundation (grant no. 2006108), which partially supported this research.
  • © Copyright 2017 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 369 (2017), 7405-7441
  • MSC (2010): Primary 03E35, 03E55, 03E75
  • DOI: https://doi.org/10.1090/tran/6974
  • MathSciNet review: 3683113