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Selective ultrafilters and $\omega\longrightarrow(\omega)^\omega$

Author(s): Todd Eisworth
Journal: Proc. Amer. Math. Soc. 127 (1999), 3067-3071.
MSC (1991): Primary 04A20
Posted: April 23, 1999
MathSciNet review: 1600136
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Abstract | References | Similar articles | Additional information

Abstract: Mathias (Happy families, Ann. Math. Logic. 12 (1977), 59-111) proved that, assuming the existence of a Mahlo cardinal, it is consistent that CH holds and every set of reals in $L(\mathbb{R})$ is $\mathcal{U}$ -Ramsey with respect to every selective ultrafilter $\mathcal{U}$ . In this paper, we show that the large cardinal assumption cannot be weakened.

References:

1.: F. Galvin and K. Prikry. Borel sets and Ramsey's theorem. J. Symbolic Logic, 38:193-198, 1973. MR 49:2399
2.: J. Henle, A.R.D. Mathias, and W.H. Woodin. A barren extension. In C. DiPrisco, editor, Methods of Mathematical Logic (Proceedings, Caracas), volume 1130 of Springer Lecture Notes in Mathematics. Springer-Verlag, 1985. MR 87d:03141
3.: K. Kunen. Some points in $\beta\mathbb{N}$ . Math. Proc. Cambridge Phil. Soc., 80:385-398, 1976. MR 55:106
4.: A. R. D. Mathias. Happy families. Ann. Math. Logic, 12:59-111, 1977. MR 58:10462
5.: F.P. Ramsey. On a problem of formal logic. Proc. London Math. Soc., 30:264-286, 1930.
6.: J. Silver. Every analytic set is Ramsey. J. Symbolic Logic, 35:60-64, 1970. MR 48:10807
7.: R.M. Solovay. A model of set theory in which every set of reals is Lebesgue measurable. Ann. of Math., 92:1-56, 1970. MR 42:64

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Additional Information:

Todd Eisworth
Affiliation: Institute of Mathematics, The Hebrew Univeristy, Jerusalem, Israel
Email: eisworth@math.huji.ac.il

DOI: 10.1090/S0002-9939-99-04835-2
PII: S 0002-9939(99)04835-2
Keywords: Selective ultrafilters, Ramsey's theorem, Mahlo cardinals
Received by editor(s): December 29, 1995
Received by editor(s) in revised form: December 10, 1997
Posted: April 23, 1999
Additional Notes: This research is part of the author's Ph.D. dissertation written at the University of Michigan under the supervision of Professor Andreas Blass. The author would like to thank the referee for his tips on streamlining the proof.
Communicated by: Andreas R. Blass
Copyright of article: Copyright 1999, American Mathematical Society

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