## The topological Baumgartner-Hajnal theorem

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- by René Schipperus PDF
- Trans. Amer. Math. Soc.
**364**(2012), 3903-3914 Request permission

## Abstract:

Two new topological partition relations are proved. These are \begin{equation*} \omega _{1} \to (top \alpha +1)^{2}_{k}\end{equation*} and \begin{equation*} \mathbb {R} \to (top \alpha +1)^{2}_{k}\end{equation*} 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

- J. Baumgartner and A. Hajnal,
*A proof (involving Martin’s axiom) of a partition relation*, Fund. Math.**78**(1973), no. 3, 193–203. MR**319768**, DOI 10.4064/fm-78-3-193-203 - G. Fodor,
*Eine Bemerkung zur Theorie der regressiven Funktionen*, Acta Sci. Math. (Szeged)**17**(1956), 139–142 (German). MR**82450** - D. A. Martin and R. M. Solovay,
*Internal Cohen extensions*, Ann. Math. Logic**2**(1970), no. 2, 143–178. MR**270904**, DOI 10.1016/0003-4843(70)90009-4 - William Weiss,
*Partitioning topological spaces*, Mathematics of Ramsey theory, Algorithms Combin., vol. 5, Springer, Berlin, 1990, pp. 154–171. MR**1083599**, DOI 10.1007/978-3-642-72905-8_{1}1

## Additional Information

**René Schipperus**- Affiliation: 1319 15 st NW, Calgary, Alberta, Canada T2N 2B7
- Email: r.schipperus@ucalgary.ca
- Received by editor(s): January 24, 2008
- Published electronically: March 21, 2012
- © Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**364**(2012), 3903-3914 - MSC (2010): Primary 03E02; Secondary 03E55
- DOI: https://doi.org/10.1090/S0002-9947-2012-04990-7
- MathSciNet review: 2912439