Selective ultrafilters and

Author:
Todd Eisworth

Journal:
Proc. Amer. Math. Soc. **127** (1999), 3067-3071

MSC (1991):
Primary 04A20

Published electronically:
April 23, 1999

MathSciNet review:
1600136

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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 is -Ramsey with respect to every selective ultrafilter . In this paper, we show that the large cardinal assumption cannot be weakened.

**1.**Fred Galvin and Karel Prikry,*Borel sets and Ramsey’s theorem*, J. Symbolic Logic**38**(1973), 193–198. MR**0337630****2.**J. M. Henle, A. R. D. Mathias, and W. Hugh Woodin,*A barren extension*, Methods in mathematical logic (Caracas, 1983) Lecture Notes in Math., vol. 1130, Springer, Berlin, 1985, pp. 195–207. MR**799042**, 10.1007/BFb0075312**3.**Kenneth Kunen,*Some points in 𝛽𝑁*, Math. Proc. Cambridge Philos. Soc.**80**(1976), no. 3, 385–398. MR**0427070****4.**A. R. D. Mathias,*Happy families*, Ann. Math. Logic**12**(1977), no. 1, 59–111. MR**0491197****5.**F.P. Ramsey. On a problem of formal logic.*Proc. London Math. Soc.*, 30:264-286, 1930.**6.**Jack Silver,*Every analytic set is Ramsey*, J. Symbolic Logic**35**(1970), 60–64. MR**0332480****7.**Robert M. Solovay,*A model of set-theory in which every set of reals is Lebesgue measurable*, Ann. of Math. (2)**92**(1970), 1–56. MR**0265151**

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Additional Information

**Todd Eisworth**

Affiliation:
Institute of Mathematics, The Hebrew Univeristy, Jerusalem, Israel

Email:
eisworth@math.huji.ac.il

DOI:
https://doi.org/10.1090/S0002-9939-99-04835-2

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

Published electronically:
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

Article copyright:
© Copyright 1999
American Mathematical Society