MORE CONSISTENCY RESULTS IN PARTITION CALCULUS

This paper has two principal aims. The first is to supply a proof of T heorem 6 of [ShSt1]: THEOREM: If ZFC+ ''there are c+ measurable cardin als'' is consistent, then so is ZFC+'' aleph(c+) is not a strong limit cardinal and aleph(c+) --> (aleph(c+), aleph1)2'' This, is done in se ctions 1 and 2, See the introduction for a discussion of the evolution of the proof and Of some interesting questions which remain open, rel ated to obstacles encountered in obtaining maximum freedom in arrangin g for any desired cardinal exponentiation in Theorems 4 and 6 of [ShSt 1]. The method is quite generally applicable in partition calculus and variants of it have in fact been applied in recent work of the author s, see [ShSt2]. First, a preservation result is proved for the game-th eoretic properties of the filters considered in [ShSt1]. Then, it is s hown that the existence of a system of such filters yields a canonizat ion-style result. Finally, it is shown that the canonization property gives the positive partition relation. The second aim makes the title of this paper slightly inaccurate (but we suspect this will be pardone d): we supply a (straightforward) proof of a result which shows the Th eorem 2 of [ShSt1] in some sense is best possible. This is done in sec tion 3.