Open covers and the square bracket partition relation
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- by Marion Scheepers PDF
- Proc. Amer. Math. Soc. 125 (1997), 2719-2724 Request permission
Abstract:
An open cover $\mathcal {U}$ of an infinite separable metric space $X$ is an $\omega$-cover of $X$ if $X\not \in \mathcal {U}$ and for every finite subset $F$ of $X$ there is a $U\in \mathcal {U}$ such that $F\subseteq U$. Let $\Omega$ be the collection of $\omega$–covers of $X$. We show that the partition relation $\Omega \rightarrow [\Omega ]^2_2$ holds if, and only if, the partition relation $\Omega \rightarrow [\Omega ]^2_3$ holds.References
- James E. Baumgartner and Alan D. Taylor, Partition theorems and ultrafilters, Trans. Amer. Math. Soc. 241 (1978), 283–309. MR 491193, DOI 10.1090/S0002-9947-1978-0491193-1
- Andreas Blass, Ultrafilter mappings and their Dedekind cuts, Trans. Amer. Math. Soc. 188 (1974), 327–340. MR 351822, DOI 10.1090/S0002-9947-1974-0351822-6
- David Booth, Ultrafilters on a countable set, Ann. Math. Logic 2 (1970/71), no. 1, 1–24. MR 277371, DOI 10.1016/0003-4843(70)90005-7
- D. Devlin, Some partition theorems and ultrafilters on $\omega$, Ph. D. thesis, Dartmouth College (1979)
- P. Erdős, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196. MR 202613, DOI 10.1007/BF01886396
- W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki, Combinatorics of Open Covers II, Topology and its Applications 73 (1996), 241–266.
- F.P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society 30 (1930), 264 – 286.
- F. Rothberger, Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae 30 (1938), 50 – 55.
- M. Sheepers, Combinatorics of Open Covers I: Ramsey theory, Topology and its Applications 69 (1996), 31–62.
- M. Scheepers, Rothberger’s property and partition relations, The Journal of Symbolic Logic, to appear.
- N.H. Williams, Combinatorial Set Theory, North-Holland (1980).
Additional Information
- Marion Scheepers
- Affiliation: Department of Mathematics and Computer Science Boise State University Boise, Idaho 83725
- MR Author ID: 293243
- Email: marion@math.idbsu.edu
- Received by editor(s): October 13, 1995
- Received by editor(s) in revised form: April 11, 1996
- Additional Notes: The author was supported by NSF grant DMS 95 - 05375
- Communicated by: Andreas R. Blass
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2719-2724
- MSC (1991): Primary 03E05
- DOI: https://doi.org/10.1090/S0002-9939-97-03898-7
- MathSciNet review: 1396995