Determinateness and Partitions

It is proved that the axiom of determinateness of Mycielski and Steinhaus for games in which players alternate in writing reals implies that co -» (co)" (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). For every infinite N C co, fi(A/) denotes the family of infinite subsets of N. We also set fi(co) = fi. A set & C fi is said to be Ramsey if there exists N G fi such that Q(N) C & otQ(N) C8-1 It is easily shown that the existence of a non-Ramsey & C fi follows from the axiom of choice. But Mathias [3] has proved that in Solovay's model [7] in which every set of reals is Lebesgue measurable and has the property of Baire, every set (J C fi is Ramsey. Silver [6] has proved that every analytic 6£ C fi is Ramsey. Ellentuck [1] has simplified Silver's proof and demonstrated that all sets in the least class containing all Borel sets and closed under the Suslin operation and complementation are Ramsey. We shall show that a form of determinateness considered by Mycielski [4] implies that every & C fi is Ramsey. This will follow fairly easily from the results of Ellentuch [1] and Oxtoby [5]. We shall start by formulating the axiom A**, due to Mycielski [4], asserting the determinateness of a class of games introduced by him. Let 7? denote the set of reals and 7?w the set of all co-sequences of reals. Let & Q R". The game &£*(($.) is played by players I and II who alternate in building an a G 7?" in the following way: I picks a0, . . . , an< (nx > 0); II picks an +,,..., an (n2 > nx); I picks «„2+1, . . . , a„3, etc. I wins if a G 6£ and II wins otherwise. A%* is the assertion that for every & C 7?" G**(&) is determined. This contradicts the axiom of choice (see [4]; this also follows from the Theorem below). Our main result is Theorem. ZF + A%* ievery & C fi is Ramsey. We shall need some results of Ellentuck [1]. An exposition of these results follows. We consider fi as a topological space with the topology defined in [1]. In order to describe this topology, we set for A G fi and n G to Received by the editors February 27, 1974 and, in revised form, May 29, 1975. AMS (MOS) subject classifications (1970). Primary 04A20, 02K05. 'The preparation of this paper, during its various stages, was supported by NSF grant GP-43841, a Fellowship from the Institute for Advanced Study and a grant from the Science Research Council. © American Mathematical Society 1976 303 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use