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Transactions of the American Mathematical Society

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Ultrafilters on $\omega$-their ideals and their cardinal characteristics

Authors: Saharon Shelah, Jörg Brendle and Saharon Shelah
Journal: Trans. Amer. Math. Soc. 351 (1999), 2643-2674
MSC (1991): Primary 03E05, 03E35
Published electronically: March 8, 1999
MathSciNet review: 1686797
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Abstract: For a free ultrafilter $\mathcal {U}$ on $\omega$ we study several cardinal characteristics which describe part of the combinatorial structure of $\mathcal {U}$. We provide various consistency results; e.g. we show how to force simultaneously many characters and many $\pi$–characters. We also investigate two ideals on the Baire space $\omega ^{\omega }$ naturally related to $\mathcal {U}$ and calculate cardinal coefficients of these ideals in terms of cardinal characteristics of the underlying ultrafilter.

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Additional Information

Jörg Brendle
Affiliation: Department of Mathematics, Dartmouth College, Bradley Hall, Hanover, New Hampshire 03755
Address at time of publication: Graduate School of Science and Technology, Kobe University, Rokko–dai, Nada, Kobe 657-8501, Japan

Saharon Shelah
Affiliation: Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel; Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Keywords: Ultrafilter, $P$–point, Ramsey ultrafilter, character, $\pi$–character, ideal, Ramsey null sets (nowhere Ramsey sets), cardinal coefficient, Mathias forcing, Laver forcing, Easton forcing
Received by editor(s): March 10, 1997
Received by editor(s) in revised form: November 4, 1997
Published electronically: March 8, 1999
Additional Notes: The research of the first author was partially supported by DFG–grant Nr. Br 1420/1–1.
The research of the second author was supported by the German–Israeli Foundation for Scientific Research & Development Grant No. G-294.081.06/93. Publication 642 on the second author’s list of publications.
Article copyright: © Copyright 1999 American Mathematical Society