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Transactions of the American Mathematical Society

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The topological Baumgartner-Hajnal theorem

Author: René Schipperus
Journal: Trans. Amer. Math. Soc. 364 (2012), 3903-3914
MSC (2010): Primary 03E02; Secondary 03E55
Published electronically: March 21, 2012
MathSciNet review: 2912439
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Abstract: Two new topological partition relations are proved. These are

$\displaystyle \omega _{1} \rightarrow (top \ \alpha +1)^{2}_{k}$    


$\displaystyle \mathbb{R} \rightarrow (top \ \alpha +1)^{2}_{k}$    

for all $ \alpha < \omega _{1}$ and all $ k< \omega $. Here the prefix ``top'' means that the homogeneous set $ \alpha +1$ is closed in the order topology. In particular, the latter relation says that if the pairs of real numbers are partitioned into a finite number of classes, there is a homogeneous (all pairs in the same class), well-ordered subset of arbitrarily large countable order type which is closed in the usual topology of the reals. These relations confirm conjectures of Richard Laver and William Weiss, respectively. They are a strengthening of the classical Baumgartner-Hajnal theorem.

References [Enhancements On Off] (What's this?)

  • [1] J. Baumgartner, A.Hajnal, A Proof (involving Martin's axiom) of a partition relation, Fund. Math. 78 no. 3 (1973), 193-203. MR 0319768 (47:8310)
  • [2] G. Fordor, Eine Bemerkung zur Theore der Regressiven Funktionen, Acta Sci. Math. Szeged 17, 139-142. MR 0082450 (18:551d)
  • [3] D.A. Martin, R.M. Solovay, Internal Cohen extensions., Ann. Math. Logic 2 no. 2 (1970), 143-178. MR 0270904 (42:5787)
  • [4] W. Weiss, Partitioning Topological Spaces in Mathematics of Ramsey Theory, Mathematics of Ramsey Theory, V. Nesetril and V. Rodel (eds.), Springer-Verlag, Heidelberg, 1990, pp. 154-171. MR 1083599

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Additional Information

René Schipperus
Affiliation: 1319 15 st NW, Calgary, Alberta, Canada T2N 2B7

Received by editor(s): January 24, 2008
Published electronically: March 21, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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