Van der Waerden spaces
Author:
Menachem Kojman
Journal:
Proc. Amer. Math. Soc. 130 (2002), 631635
MSC (2000):
Primary 05C55, 54F65; Secondary 03E05, 11P99, 26A48
Published electronically:
August 28, 2001
MathSciNet review:
1866012
Fulltext PDF Free Access
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 firstcountable.
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., WileyInterscience Series
in Discrete Mathematics and Optimization, John Wiley & Sons Inc., New
York, 1990. A WileyInterscience Publication. MR 1044995
(90m:05003)
 [2]
Neil
Hindman and Dona
Strauss, Algebra in the StoneČech compactification, de
Gruyter Expositions in Mathematics, vol. 27, Walter de Gruyter &
Co., Berlin, 1998. Theory and applications. MR 1642231
(99j:54001)
 [3]
Lynn
Arthur Steen and J.
Arthur Seebach Jr., Counterexamples in topology, 2nd ed.,
SpringerVerlag, New York, 1978. MR 507446
(80a:54001)
 [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
(98g:28017), http://dx.doi.org/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 (90k:03046), http://dx.doi.org/10.1090/S00029947199009822325
 [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 (50 #2067)
 [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
(81m:10019)
 [8]
Neil
Hindman, Partitions and sums and products of
integers, Trans. Amer. Math. Soc. 247 (1979), 227–245. MR 517693
(80b:10022), http://dx.doi.org/10.1090/S00029947197905176934
 [9]
B. L. van der Waerden.
Beweis eine Baudetschen Vermutung Nieus Arch. Wisk., 15:212216, 1927.
 [1]
 Ronald L. Graham, Bruce L. Rothschild and Joel H. Spencer.
Ramsey Theory, John Wiley & Sons, 1990. MR 90m:05003
 [2]
 N. Hindman and D. Strauss.
Algebra in the StoneCech compactification, de Gruyter Berlin, 1998. MR 99j:54001
 [3]
 Lynn Arthur Steen and J. Arthur Seebach Jr.,
Counterexamples in topology, SpringerVerlag, New York, 1978. MR 80a:54001
 [4]
 V. Bergelson.
Ergodic Ramsey Theory  an update. Ergodic theory and actions. London Math. Soc. Lecture Note series 228, 161, 1996. MR 98g:28017
 [5]
 V. Bergelson and N. Hindman.
Nonmetrizable topological dynamics and Ramsey Theory. Trans. Amer. Math. Soc., 320:293320, 1990. MR 90k:03046
 [6]
 N. Hindman.
Finite sums from sequences within cells of partitions of . J. Combinatorial Theory Ser. A, 17:119184, 1974. MR 50:2067
 [7]
 N. Hindman.
Ultrafilters and combinatorial number theory. Number Theory, Carbondale, M. Nathanson, ed. Lecture Notes in Math., 751, 1979. MR 81m:10019
 [8]
 N. Hindman.
Partitions and sums and products of integers. Trans. Amer. Math. Soc., 247:227245, 1979. MR 80b:10022
 [9]
 B. L. van der Waerden.
Beweis eine Baudetschen Vermutung Nieus Arch. Wisk., 15:212216, 1927.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
05C55,
54F65,
03E05,
11P99,
26A48
Retrieve articles in all journals
with MSC (2000):
05C55,
54F65,
03E05,
11P99,
26A48
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/S0002993901061160
PII:
S 00029939(01)061160
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
