Effectiveness for the Dual Ramsey Theorem

The Determined Property of Baire in Reverse Math

“…If we represent the coloring in the natural way described below, then DPB can be used to carry out (1). It was known to Simpson (see [DFSW17]) that CDRT 3 ℓ follows from Hindman's Theorem (HT), which follows from ACA + 0 by [BHS87]. Therefore, the following natural formalization of Borel-DRT 3 ℓ follows from DPB + ACA + 0 .…”

“…Therefore, the Borel dual Ramsey theorem has a natural formulation in terms of determined Borel sets. In [PV85,DFSW17], it was shown that a solution to Borel-DRT k ℓ can in general be obtained by a two-step process: (1) Use the fact that every Borel set has the property of Baire to come up with a Baire approximation to each color in the given coloring.…”

“…Even when ATR 0 is not taken as the base theory, theorems involving Borel sets tend to imply ATR 0 . The probable reason for this was observed in [DFSW17].…”

“…Our motivation is to make it possible to give a meaningful reverse math analysis of theorems whose statements involve Borel sets. The way that Borel sets are usually defined in reverse math forces many theorems that even mention a Borel set to imply ATR 0 , in an unsatisfactory sense made precise in [DFSW17]. Here we propose another definition for a Borel set in reverse math, distinguished from the original by the terminology determined Borel set, and to put bounds on the strength of the statement DPB : "Every determined Borel set has the property of Baire" This statement should be compared with the usual "Every Borel set has the property of Baire", which [DFSW17] showed is equivalent to ATR 0 for aforementioned empty reasons.…”

“…A basic proposition is that every Borel set has the property of Baire, but what is the strength of that proposition in reverse math? In [DFSW17], the relevant notions were formalized as follows.…”