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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Canonical forms of Borel functions on the Milliken space

Author(s): Olaf Klein; Otmar Spinas
Journal: Trans. Amer. Math. Soc. 357 (2005), 4739-4769.
MSC (2000): Primary 03E15, 05D10, 54H05
Posted: July 19, 2005
MathSciNet review: 2165386
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Abstract | References | Similar articles | Additional information

Abstract: The goal of this paper is to canonize Borel measurable mappings $\Delta\colon\Omega^\omega\to\mathbb{R} $, where $\Omega^\omega$ is the Milliken space, i.e., the space of all increasing infinite sequences of pairwise disjoint nonempty finite sets of $\omega$. This main result is a common generalization of a theorem of Taylor and a theorem of Prömel and Voigt.

References:

[Ra30]
F. P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society (2), vol. 30 (1930), pp. 264-286.

[ErRa50]
P. Erdös and R. Rado, A combinatorial theorem, J. London Math. Soc., vol. 25 (1950), pp. 249-255. MR 0037886 (12:322f)

[Ku66]
K. Kuratowski, Topology, Academic Press, New York, vol. 1 (1966). MR 0217751 (36:840)

[Ma68]
A. R. D. Mathias, On a generalization of Ramsey's theorems, Notices of the American Mathematical Society, vol. 15 (1968), p. 931.

[Si70]
J. Silver, Every analytic set is Ramsey, The Journal of Symbolic Logic, vol. 35 (1970), no. 1, pp. 60-64. MR 0332480 (48:10807)

[GrRo71]
R. L. Graham and B. L. Rothschild, Ramsey's theorem for $n$-parameter sets, Trans. Amer. Math. Soc., vol. 159 (1971), pp. 413-432. MR 0284352 (44:1580)

[GaPr73]
F. Galvin and K. Prikry, Borel sets and Ramsey's Theorem, The Journal of Symbolic Logic, vol. 38 (1973), no. 2, pp. 193-198. MR 0337630 (49:2399)

[Ba74]
J. E. Baumgartner, A short proof of Hindman's Theorem, Journal of Combinatorial Theory (A), vol. 17 (1974), pp. 384-386. MR 0354394 (50:6873)

[El74]
E. Ellentuck, A new proof that analytic sets are Ramsey, The Journal of Symbolic Logic, vol. 39 (1974), no. 1, pp. 163-165. MR 0349393 (50:1887)

[Hi74]
N. Hindman, Finite sums from sequences within cells of a partition of $N$, Journal of Combinatorial Theory (A), vol. 17 (1974), pp. 1-11. MR 0349574 (50:2067)

[Mi75]
K. R. Milliken, Ramsey's Theorem with Sums or Unions, Journal of Combinatorial Theory (A), vol. 18 (1975), pp. 276-290. MR 0373906 (51:10106)

[Ta76]
A. D. Taylor, A Canonical Partition Relation for Finite Subsets of $\omega$, Journal of Combinatorial Theory (A), vol. 21 (1976), pp. 137-146. MR 0424571 (54:12530)

[Ma77]
A. R. D. Mathias, Happy families, Annals of Math. Logic, vol. 12 (1977), pp. 59-111. MR 0491197 (58:10462)

[PuRö82]
P. Pudlak and V. Rödl, Partition theorems for systems of finite subsets of integers, Discrete Math., vol. 39 (1982), pp. 67-73. MR 0677888 (84i:05018)

[PrVo85]
H. J. Prömel and B. Voigt, Canonical Forms of Borel-Measurable Mappings $\Delta\colon[\omega]^\omega\to\mathbb{R} $, Journal of Combinatorial Theory (A), vol. 40 (1985), pp. 409-417. MR 0814423 (87g:04004)

[Ke95]
A. S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, Berlin, 1995. MR 1321597 (96e:03057)

[To98]
S. Todorcevic, Infinite-Dimensional Ramsey Theory, Preprint, (1998).

[Sp01]
O. Spinas, Canonical behaviour of Borel functions on superperfect rectangles, Journal of Math. Logic, vol. 1, no. 2 (2001), pp. 173-220. MR 1864736 (2002h:03107)

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Additional Information:

Olaf Klein
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität Zu Kiel, Ludewig-Meyn-Strasse 4, 24098 Kiel, Germany

Otmar Spinas
Affiliation: Mathematisches Seminar, Christian-Albrechts-Universität Zu Kiel, Ludewig-Meyn-Strasse 4, 24098 Kiel, Germany
Email: spinas@math.uni-kiel.de

DOI: 10.1090/S0002-9947-05-04000-6
PII: S 0002-9947(05)04000-6
Received by editor(s): March 12, 2002
Posted: July 19, 2005
Additional Notes: The second author was partially supported by DFG grant SP 683
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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