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We further develop the theory of layered semigroups, as introduced by Farah, Hindman and McLeod, providing a general framework to prove Ramsey statements about such a semigroup S. By nonstandard and topological arguments, we show Ramsey statements on S are implied by the existence of "coherent" sequences in S. This framework allows us to formalise and prove many results in Ramsey theory, including Gowers' FIN k theorem, the Graham-Rothschild theorem, and Hindman's finite sums theorem. Other highlights include: a simple nonstandard proof of the Graham-Rothschild theorem for strong variable words; a nonstandard proof of Bergelson-Blass-Hindman's partition theorem for located variable words, using a result of Carlson, Hindman and Strauss; and a common generalisation of the latter result and Gowers' theorem, which can be proven in our framework.Dedicated to the late Ronald L. Graham (1935Graham ( -2020. This paper would never have been written if not for his groundbreaking work in Ramsey theory.

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<p>Using the tools of reverse mathematics in second-order arithmetic, as developed by Friedman, Simpson, and others, we determine the axioms necessary to develop various topics in commutative ring theory. Our main contributions to the field are as follows. We look at fundamental results concerning primary ideals and the radical of an ideal, concepts previously unstudied in reverse mathematics. Then we turn to a fine-grained analysis of four different definitions of Noetherian in the weak base system RCA_0 + Sigma-2 induction. Finally, we begin a systematic study of various types of integral domains: PIDs, UFDs and Bézout and GCD domains.</p>