Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [Formula: see text] principles over [Formula: see text]-models of [Formula: see text]. They also introduced a version of this game that similarly captures provability over [Formula: see text]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [Formula: see text] between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between [Formula: see text] principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is “purely proof-theoretic”, as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of [Formula: see text], uncovering new differences between their logical strengths.

This paper is part of a line of research on the computability-theoretic and reverse-mathematical strength of versions of Hindman's Theorem [6] that began with the work of Blass, Hirst, and Simpson [1], and has seen considerable interest recently. We assume basic familiarity with computability theory and reverse mathematics, at the level of the background material in [8], for instance. On the reverse mathematics side, the two major systems with which we will be concerned are RCA 0 , the usual weak base system for reverse mathematics, which corresponds roughly to computable mathematics; and ACA 0 , which corresponds roughly to arithmetic mathematics. For principles P of the form (∀X) [Φ(X) → (∃Y ) Ψ(X, Y )], we call any X such that Φ(X) holds an instance of P , and any Y such that Ψ(X, Y ) holds a solution to X.We begin by introducing some related combinatorial principles. For a set S, let [S] n be the set of n-element subsets of S. Ramsey's Theorem (RT) is the statement that for every n and every coloring of [N] n with finitely many colors, there is an infinite set H that is homogeneous for c, which means that all elements of [H] n have the same color. There has been a great deal of work on computability-theoretic and reverse-mathematical aspects of versions of Ramsey's Theorem, such as RT n k , which is RT restricted to colorings of [N] n with k many colors. (See e.g. [8].

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Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between Π 1 2 principles over ω-models of RCA 0 . They also introduced a version of this game that similarly captures provability over RCA 0 . We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication Q → P between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles.We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computabilitytheoretic notions such as Weihrauch reducibility, allowing for a kind