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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Van der Waerden spaces and Hindman spaces are not the same


Authors: Menachem Kojman and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 131 (2003), 1619-1622
MSC (2000): Primary 54A20, 05A17, 03E35; Secondary 03E50
Published electronically: December 16, 2002
MathSciNet review: 1950294
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Abstract: A Hausdorff topological space $X$ is van der Waerden if for every sequence $(x_n)_{n\in\omega}$ in $X$ there is a converging subsequence $(x_n)_{n\in A}$ where $A\subseteq \omega$ contains arithmetic progressions of all finite lengths. A Hausdorff topological space $X$ is Hindman if for every sequence $(x_n)_{n\in\omega}$ in $X$there is an IP-converging subsequence $(x_n)_{n\in FS(B)}$for some infinite $B\subseteq\omega$.

We show that the continuum hypothesis implies the existence of a van der Waerden space which is not Hindman.


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Additional Information

Menachem Kojman
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel
Email: kojman@cs.bgu.ac.il

Saharon Shelah
Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
Email: shelah@ma.huji.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06916-2
PII: S 0002-9939(02)06916-2
Received by editor(s): September 13, 2001
Received by editor(s) in revised form: December 12, 2001
Published electronically: December 16, 2002
Additional Notes: The first author was partially supported by an Israel Science Foundation grant
The second author was partially supported by an Israel Science Foundation grant. Number 782 in Shelah’s list of publications.
The authors wish to acknowledge a substantial simplification made by the referee in the proof. The referee has eliminated an inessential use that the authors have made of the canonical van der Waerden theorem, all of whose known proofs use Szemerédi’s theorem.
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society