ω-reducibility of BRT^3_2 to RT^2_2

Determine whether bounded Ramsey's theorem for 2-colorings of triples (BRT^3_2) is ω-reducible to Ramsey's theorem for 2-colorings of pairs (RT^2_2); equivalently, ascertain whether every ω-model of RCA_0 + RT^2_2 is an ω-model of BRT^3_2.

Background

The paper studies BRT^3_2, the bounded version of Ramsey’s theorem for triples with two colors, and compares its computability-theoretic strength with that of RT^2_2, Ramsey’s theorem for pairs with two colors. The authors show that BRT^3_2 satisfies many of the same upper bounds known for RT^2_2 (e.g., cone avoidance, constant-bound trace avoidance, and a weakly low basis).

Despite these similarities, they also prove separations indicating differences between the principles: BRT^3_{2,4} is not computably reducible to RT^2, and for every fixed k, BRT^3_{2,4} is not ω-reducible to RT^2_2 in k or fewer moves. However, the full question of ω-reducibility, namely whether BRT^3_2 ≤_ω RT^2_2, remains unresolved and is explicitly left open.