Implications between Milliken’s tree theorem, the tree theorem, and Weak König’s Lemma

Determine whether Milliken’s tree theorem for pairs (MTT2) implies Weak König’s Lemma (WKL0) over RCA0, and whether the Chubb–Hirst–McNicholl tree theorem for pairs (TT2) implies Milliken’s tree theorem for pairs (MTT2) over RCA0.

Background

The paper situates the Ginsburg–Sands theorem within the reverse mathematics landscape and compares it to other combinatorial principles lying strictly between ACA0 and RT2. In this context, it recalls known separations among tree-based theorems and notes gaps that remain.

Specifically, although several non-implications have been established (e.g., MTT2 does not imply ACA0; TT2 does not imply ACA0 or even WKL0; RT2 does not imply TT2), the relationships MTT2 ⇒ WKL0 and TT2 ⇒ MTT2 have not been resolved.