Van der Waerden spaces and Hindman spaces are not the same
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- by Menachem Kojman and Saharon Shelah PDF
- Proc. Amer. Math. Soc. 131 (2003), 1619-1622 Request permission
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.References
- Neil Hindman, Finite sums from sequences within cells of a partition of $N$, J. Combinatorial Theory Ser. A 17 (1974), 1–11. MR 349574, DOI 10.1016/0097-3165(74)90023-5
- Menachem Kojman, van der Waerden spaces, Proc. Amer. Math. Soc. 130 (2002), no. 3, 631–635. MR 1866012, DOI 10.1090/S0002-9939-01-06116-0
- M. Kojman. Hindman spaces. Proc. Amer. Math. Soc., 130:1597–1602, 2002.
- B. L. van der Waerden. Beweis eine Baudetschen Vermutung Nieuw Arch. Wisk., 15:212–216, 1927.
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
- MR Author ID: 160185
- ORCID: 0000-0003-0462-3152
- Email: shelah@ma.huji.ac.il
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1619-1622
- MSC (2000): Primary 54A20, 05A17, 03E35; Secondary 03E50
- DOI: https://doi.org/10.1090/S0002-9939-02-06916-2
- MathSciNet review: 1950294