LARGE TURING INDEPENDENT SETS

For a set of reals X and 1 ≤ n < ω, define X to be n-Turing independent iff the Turing join of any n reals in X does not compute another real in X. X is Turing independent iff it is n-Turing independent for every n. We show the following: (1) There is a non-meager Turing independent set. (2) The statement “Every set of reals of size continuum has a Turing independent subset of size continuum” is independent of ZFC plus the negation of CH. (3) The statement “Every non-meager set of reals has a non-meager n-Turing independent subset” holds in ZFC for n = 1 and is independent of ZFC for n ≥ 2 (assuming the consistency of a measurable cardinal). We also show the measure analogue of (3).