Van der Waerden spaces
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- by Menachem Kojman PDF
- Proc. Amer. Math. Soc. 130 (2002), 631-635 Request permission
Abstract:
A topological space $X$ is van der Waerden if for every sequence $(x_n)_n$ in $X$ there exists a converging subsequence $(x_{n_k})_k$ so that $\{{n_k}:k\in \mathbb {N}\}$ contains arbitrarily long finite arithmetic progressions. Not every sequentially compact space is van der Waerden. The product of two van der Waerden spaces is van der Waerden.
The following condition on a Hausdorff space $X$ is sufficent for $X$ to be van der Waerden:
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[$(*)$] The closure of every countable set in $X$ is compact and first-countable.
A Hausdorff space $X$ that satisfies $(*)$ satisfies, in fact, a stronger property: for every sequence $(x_n)$ in $X$:
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[$(\star )$] There exists $A\subseteq \mathbb {N}$ so that $(x_n)_{n\in A}$ is converging, and $A$ contains arbitrarily long finite arithmetic progressions and sets of the form $FS(D)$ for arbitrarily large finite sets $D$.
There are nonmetrizable and noncompact spaces which satisfy $(*)$. In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on $[0,1]$ satisfy $(\star )$.
References
- Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, 2nd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR 1044995
- Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification, De Gruyter Expositions in Mathematics, vol. 27, Walter de Gruyter & Co., Berlin, 1998. Theory and applications. MR 1642231, DOI 10.1515/9783110809220
- Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR 507446
- Vitaly Bergelson, Ergodic Ramsey theory—an update, Ergodic theory of $\textbf {Z}^d$ actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 1–61. MR 1411215, DOI 10.1017/CBO9780511662812.002
- Vitaly Bergelson and Neil Hindman, Nonmetrizable topological dynamics and Ramsey theory, Trans. Amer. Math. Soc. 320 (1990), no. 1, 293–320. MR 982232, DOI 10.1090/S0002-9947-1990-0982232-5
- 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
- Neil Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 119–184. MR 564927
- Neil Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979), 227–245. MR 517693, DOI 10.1090/S0002-9947-1979-0517693-4
- B. L. van der Waerden. Beweis eine Baudetschen Vermutung Nieus Arch. Wisk., 15:212–216, 1927.
Additional Information
- Menachem Kojman
- Affiliation: Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva 84105, Israel
- Email: kojman@math.bgu.ac.il
- Received by editor(s): August 2, 2000
- Received by editor(s) in revised form: August 28, 2000
- Published electronically: August 28, 2001
- Additional Notes: The author thanks Uri Abraham for many discussions that contributed to the development of this paper, and also thanks the referee for some constructive and helpful comments
- Communicated by: Alan Dow
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 631-635
- MSC (2000): Primary 05C55, 54F65; Secondary 03E05, 11P99, 26A48
- DOI: https://doi.org/10.1090/S0002-9939-01-06116-0
- MathSciNet review: 1866012