First-jump control for T1 CSC spaces

Investigate whether, for every set P not of PA degree and every infinite computable T1 countable second-countable (CSC) space X,U,k, there exists an infinite subspace Y that is either discrete or has the cofinite topology and whose Turing jump is bounded by P′, i.e., Y′ ≤T P′; the authors conjecture the answer is negative.

Background

The paper draws an analogy to the 'first jump control' technique for RT2 (as in Cholak–Jockusch–Slaman), where given P ≫ ∅ and a computable coloring c, one can find a homogeneous set H with H ≤T P′.

Translating this technique to the setting of GST1 (the T1 case of the Ginsburg–Sands theorem) would produce subspaces with controlled jumps and yield further separations below RT2. The authors, however, suspect this control fails in the T1 CSC setting.