Determinateness and partitions

Author:
Karel Prikry

Journal:
Proc. Amer. Math. Soc. **54** (1976), 303-306

MSC:
Primary 04A20; Secondary 04A25

MathSciNet review:
0453540

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Abstract: It is proved that the axiom of determinateness of Mycielski and Steinhaus for games in which players alternate in writing reals implies that (i.e. for every partition of infinite sets of natural numbers into two classes there is an infinite set such that all its infinite subsets belong to the same class).

**[1]**Erik Ellentuck,*A new proof that analytic sets are Ramsey*, J. Symbolic Logic**39**(1974), 163–165. MR**0349393****[2]**Fred Galvin and Karel Prikry,*Borel sets and Ramsey’s theorem*, J. Symbolic Logic**38**(1973), 193–198. MR**0337630****[3]**A. R. D. Mathias,*On a generalization of Ramsey's theorem*, Thesis, Bonn, 1968.**[4]**Jan Mycielski,*On the axiom of determinateness. II*, Fund. Math.**59**(1966), 203–212. MR**0210603****[5]**John C. Oxtoby,*The Banach-Mazur game and Banach category theorem*, Contributions to the theory of games, vol. 3, Annals of Mathematics Studies, no. 39, Princeton University Press, Princeton, N. J., 1957, pp. 159–163. MR**0093741****[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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DOI:
https://doi.org/10.1090/S0002-9939-1976-0453540-X

Article copyright:
© Copyright 1976
American Mathematical Society