Several notions of computability-theoretic reducibility between [Formula: see text] principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey’s Theorem and related principles under these notions. Among other results, we show that for each [Formula: see text], there is an instance of RT[Formula: see text] all of whose solutions have PA degree over [Formula: see text] and use this to show that König’s Lemma lies strictly between RT[Formula: see text] and RT[Formula: see (...) text] under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer on comparing versions of Ramsey’s Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey-theoretic properties, we show that for each [Formula: see text], there is an [Formula: see text]-coloring [Formula: see text] of [Formula: see text] such that every [Formula: see text]-coloring of [Formula: see text] has an infinite homogeneous set that does not compute any infinite homogeneous set for [Formula: see text], and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to [Formula: see text]-models of RCA0, and related notions that allow us to count how many applications of a principle [Formula: see text] are needed to reduce another principle to [Formula: see text]. Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman. (shrink)

Ramsey’s theorem states that for any coloring of then-element subsets of ℕ with finitely many colors, there is an infinite setHsuch that alln-element subsets ofHhave the same color. The strength of consequences of Ramsey’s theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey’s theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose (...) during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey’s theorem. (shrink)