The principle ADS asserts that every linear order on ω has an infinite ascending or descending sequence. This has been studied extensively in the reverse mathematics literature, beginning with the work of Hirschfeldt and Shore [16]. We introduce the principle ADC, which asserts that every such linear order has an infinite ascending or descending chain. The two are easily seen to be equivalent over the base system RCA 0 of second order arithmetic; they are even computably equivalent. However, we prove that ADC is strictly weaker than ADS under Weihrauch (uniform) reducibility. In fact, we show that even the principle SADS, which is the restriction of ADS to linear orders of type ω + ω * , is not Weihrauch reducible to ADC. In this connection, we define a more natural stable form of ADS that we call General-SADS, which is the restriction of ADS to linear orders of type k + ω, ω + ω * , or ω + k, where k is a finite number. We define General-SADC analogously. We prove that General-SADC is not Weihrauch reducible to SADS, and so in particular, each of SADS and SADC is strictly weaker under Weihrauch reducibility than its general version. Finally, we turn to the principle CAC, which asserts that every partial order on ω has an infinite chain or antichain. This has two previously studied stable variants, SCAC and WSCAC, which were introduced by Hirschfeldt and Jockusch [16], and by Jockusch, Kastermans, Lempp, Lerman, and Solomon [18], respectively, and which are known to be equivalent over RCA 0 . Here, we show that SCAC is strictly weaker than WSCAC under even computable reducibility.

Abstract. The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to nonuniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q implies a theorem P, it is usually because there is a direct uniform translation of the problems represented by P into the problems represented by Q, in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all n, j, k ≥ 1, if j < k then Ramsey's theorem for n-tuples and k many colors is not uniformly, or Weihrauch, reducible to Ramsey's theorem for n-tuples and j many colors. The two theorems are classically equivalent, so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions. We also study Weak König's Lemma, the Thin Set Theorem, and the Rainbow Ramsey's Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve infinitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related.

Abstract. We introduce and study several notions of computability-theoretic reducibility between subsets of ω that are "robust" in the sense that if only partial information is available about the oracle, then partial information can be recovered about the output. These are motivated by reductions between Π 1 2 principles in the context of reverse mathematics, and also encompasses generic and coarse reducibilities, previously studied by Jockusch and Schupp [JS-2012].

Abstract. We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let HT ≤n k denote the assertion that for each k-coloring c of N there is an infinite set X ⊆ N such that all sums x∈F x for F ⊆ X and 0 < |F | ≤ n have the same color. We prove that there is a computable 2-coloring c of N such that there is no infinite computable set X such that all nonempty sums of at most 2 elements of X have the same color. It follows that HT ≤2 2 is not provable in RCA 0 and in fact we show that it implies SRT 2 2 in RCA 0 . We also show that there is a computable instance of HT ≤3 3 with all solutions computing 0 ′ . The proof of this result shows that HT ≤3 3 implies ACA 0 in RCA 0 .

Hindman's Theorem (HT) states that for every coloring of N with finitely many colors, there is an infinite set H ⊆ N such that all nonempty sums of distinct elements of H have the same color. The investigation of restricted versions of HT from the computability-theoretic and reverse-mathematical perspectives has been a productive line of research recently. In particular, HT n k is the restriction of HT to sums of at most n many elements, with at most k colors allowed, and HT =n