Van der Waerden spaces

Author:
Menachem Kojman

Journal:
Proc. Amer. Math. Soc. **130** (2002), 631-635

MSC (2000):
Primary 05C55, 54F65; Secondary 03E05, 11P99, 26A48

Published electronically:
August 28, 2001

MathSciNet review:
1866012

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Abstract | References | Similar Articles | Additional Information

Abstract: A topological space is *van der Waerden* if for every sequence in there exists a converging subsequence so that 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 is sufficent for to be van der Waerden:

- The closure of every countable set in is compact and first-countable.

A Hausdorff space that satisfies satisfies, in fact, a stronger property: for every sequence in :

- There exists so that is converging, and contains arbitrarily long finite arithmetic progressions and sets of the form for arbitrarily large finite sets .

There are nonmetrizable and noncompact spaces which satisfy . In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on satisfy .

**[1]**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****[2]**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****[3]**Lynn Arthur Steen and J. Arthur Seebach Jr.,*Counterexamples in topology*, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR**507446****[4]**Vitaly Bergelson,*Ergodic Ramsey theory—an update*, Ergodic theory of 𝑍^{𝑑} actions (Warwick, 1993–1994) London Math. Soc. Lecture Note Ser., vol. 228, Cambridge Univ. Press, Cambridge, 1996, pp. 1–61. MR**1411215**, 10.1017/CBO9780511662812.002**[5]**Vitaly Bergelson and Neil Hindman,*Nonmetrizable topological dynamics and Ramsey theory*, Trans. Amer. Math. Soc.**320**(1990), no. 1, 293–320. MR**982232**, 10.1090/S0002-9947-1990-0982232-5**[6]**Neil Hindman,*Finite sums from sequences within cells of a partition of 𝑁*, J. Combinatorial Theory Ser. A**17**(1974), 1–11. MR**0349574****[7]**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****[8]**Neil Hindman,*Partitions and sums and products of integers*, Trans. Amer. Math. Soc.**247**(1979), 227–245. MR**517693**, 10.1090/S0002-9947-1979-0517693-4**[9]**B. L. van der Waerden.

Beweis eine Baudetschen Vermutung*Nieus Arch. Wisk.*, 15:212-216, 1927.

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Additional Information

**Menachem Kojman**

Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva 84105, Israel

Email:
kojman@math.bgu.ac.il

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-06116-0

Keywords:
van der Waerden's Theorem,
converging sequence,
compactification,
finite sums

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

Article copyright:
© Copyright 2001
American Mathematical Society