Δ¹₂-sets of reals

We consider and give a complete solution to, implications of the form: (*) Every X₁-set of reals has the property P₁ implies every X₂-set of reals has the property P₂, for X₁,X₂ε{lunate} {Δ¹₂,Π¹₁, σ¹₂,Π¹₂}, and where P₁,P₂ are among 'to be Ramsey', 'K_σ-regular' and of course 'Lebesgue measurable' and 'Baire categoricity'. Naturally we are led to look for characterizations of such properties (by forcing). Not surprisingly, excepting the trivial implications, we get many consistency results, but 'fortunately' we get quite a number of theorems ( = implications proved in ZFC), notably among the 'to be Ramsay' and 'K_σ-regular'. 1. Theorem 1. The following are equivalent: (a) Every Σ¹₂-set of reals is Ramsey. (b) Every Δ¹₂-set of reals is Ramsey. (c) For every r ε{lunate} R there exists s ε{lunate} [ω]^ω, s is P(D_s[r])-generic over L[r][D_s] (Definitions are given in Section 0.) For this theorem we develop a forcing P(D) (D an ultrafilter on ω) shooting a real 'through' the ultrafilter. 1. Theorem 2. The following are equivalent: (a) Every Σ¹₂-set of reals is K_σ-regular. (b) Every Δ¹₂-set of reals is K_σ-regular. (c) Every Π¹₁-set of reals is K_σ-regular. (d) For every rε{lunate}R, there exists fε{lunate} ω^ω, f is a σ-bound to ω^ω ∩L[r].