Robert Wolf

We prove (in ZFC Set Theory) that all infinite games whose winning sets are of the following forms are determined: (1) (A - S) ∪ B, where A is $\Pi^0_2, \bar\bar{S}, 2^{\aleph_0}$ , and the games whose winning set is B is "strongly determined" (meaning that all of its subgames are determined). (2) A Boolean combination of Σ 0 2 sets and sets smaller than the continuum. This also enables us to show that strong determinateness is not preserved under complementation, improving a result of Morton Davis which required the continuum hypothesis to prove this fact. Various open questions related to the above results are discussed. Our main conjecture is that (2) above remains true when "Σ 0 2 " is replaced by "Borel"