Turing independence and Baire category

We show that it is relatively consistent with ZFC that there is a non-meager set of reals X such that for every non-meager Y subset of X, there exist distinct x, y, z is an element of Y such that z is computable from the Turing join of x and y.

Automated summary

It is shown in this study that in the context of ZFC, there exists a relatively consistent scenario where a non-meager set of real numbers X exists. For every non-meager subset Y of X, there are distinct elements x, y, z in Y such that z is computable from the Turing join of x and y.