Ramsey monoids

树的自对偶拉姆齐度

Self-Dual Ramsey Degrees for Trees

该研究探讨树之间由Deuber嵌入与Solecki刚性满射构成的映射对的拉姆齐问题，证明双坐标染色下无拉姆齐定理，给出其拉姆齐度刻画，并表明该结果可推出Solecki的线性序映射对拉姆齐定理。

Note on Ramsey theorem for posets with linear extensions

In this note we consider a Ramsey type result for partially ordered sets. In particular, we give an alternative short proof of a theorem for a posets with multiple linear extensions recently obtained by Solecki and Zhao.

Almost $\mathcal{R}$-trivial monoids are almost never Ramsey

Recent results have generalized Gowers' Theorem (to Lupini's Theorem) and the Furstenberg-Katznelson theorem, both infinite dimensional Ramsey Theorems. The framework of arXiv:1611.06600 provides a machine which accepts (almost $\mathcal{R}$-trivial) monoids and outputs Ramsey theorems. The major generalization was to extract monoid actions from these theorems. We investigate the other direction, and feed into the machine monoids which appear "naturally", and which are not extracted from a Ramsey theorem, such as such as $0$-Hecke monoids and hyperplane face monoids. Most examples of monoids c…

Monoid actions and ultrafilter methods in Ramsey theory

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 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 i…

A Ramsey Theorem for Multiposets

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result of J. Nešetřil and V. Rödl claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 M. Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of S. Solecki and M. Zhao that the class of all finite posets with several linear extensions has the Ramsey property. Using the categorical reinterpretation of the Ra…

Conant's generalised metric spaces are Ramsey

We give Ramsey expansions of classes of generalised metric spaces where distances come from a linearly ordered commutative monoid. This complements results of Conant about the extension property for partial automorphisms and extends an earlier result of the first and the last author giving the Ramsey property of convexly ordered $S$-metric spaces. Unlike Conant's approach, our analysis does not require the monoid to be semi-archimedean.