Geodesic
MONOID ACTIONS AND ULTRAFILTER METHODS IN RAMSEY THEORY
Forum of Mathematics, Sigma, Tome 7 (2019)

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First, we prove a theorem on dynamics of actions of monoids by endomorphisms of semigroups. Second, we introduce algebraic structures suitable for formalizing infinitary Ramsey statements and prove a theorem that such statements are implied by the existence of appropriate homomorphisms between the algebraic structures. We make a connection between the two themes above, which allows us to prove some general Ramsey theorems for sequences. We give a new proof of the Furstenberg–Katznelson Ramsey theorem; in fact, we obtain a version of this theorem that is stronger than the original one. We answer in the negative a question of Lupini on possible extensions of Gowers’ Ramsey theorem. 
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     author = {S{\L}AWOMIR SOLECKI},
     title = {MONOID {ACTIONS} {AND} {ULTRAFILTER} {METHODS} {IN} {RAMSEY} {THEORY}},
     journal = {Forum of Mathematics, Sigma},
     publisher = {mathdoc},
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     year = {2019},
     doi = {10.1017/fms.2018.28},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2018.28/}
}
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SŁAWOMIR SOLECKI. MONOID ACTIONS AND ULTRAFILTER METHODS IN RAMSEY THEORY. Forum of Mathematics, Sigma, Tome 7 (2019). doi: 10.1017/fms.2018.28

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