There are some results in Ramsey theory, which involve additive structure of $(\mathbb N,+)$.
For example, if we color the set $\mathbb N$ by finitely many colors, then:
- There are three numbers $x$, $y$, $z$ such that $x+y=z$ and they are colored by the same color. (Schur's theorem)
- There exists a monochromatic set, which contains arbitrary long finite arithmetic progressions. (Van der Waerden's theorem)
- There exists a monochromatic set $A$ which contains an infinite subset $D$ such that $FS(D)\subseteq A$, where $$FS(D)=\{\sum_{a\in F} a; F\subseteq D, F\text{ is finite}\}$$ is the set of all (non-repetitive) sums of finite subset of $D$. (Hindman's theorem)
Similar questions can be asked if we replace $(\mathbb N,+)$ by any semigroup $(S,\cdot)$. (We replace the condition $x+y=z$ by $x\cdot y=z$. We replace the arithmetical progression $n,n+d,\dots,n+kd$ with $n,n\cdot d, \dots, n\cdot d^k$. Finite sums are replaced by $d_1\cdot d_2 \cdots d_k$ for $\{d_1,\dots,d_k\}\subseteq D$.) Although when $S$ is not commutative, these reformulations start to have a bit "different feel" to them.
Are the above results (or other Ramsey-theoretic results) valid fort some larger classes of semigroups rather then just $(\mathbb N,+)$?
Motivation for asking this question is that I recently read the proof of Hindman's theorem and van der Waerden's theorem in Todorcevic's Topics in Topology (LNM 1652, doi: 10.1007/BFb0096295). These proofs are using the Stone–Čech compactification $(\beta N,+)$. But the construction of $(\beta S,\cdot)$ can be done in the same way for any semigroup $(S,\cdot)$.
For example, the basic ingredient in the proof of Hindman's theorem was existence of idempotent ultrafilter. Such ultrafilter exists in any $\beta S$ by Auslander-Ellis-Numakura lemma. So it seems that this proof of Hindman's theorem might be valid for any infinite semigroups (or at least if it is commutative). However, it is very probable that in this short sketch I have missed some details and we need to impose some conditions on $S$ for the same proof to work.
I was not sure whether formulating this question in this way (asking about several theorems) does not result in making this question too broad. But I thought that this is better than posting three separate questions. I am not looking for detailed proofs, just for some pointers to literature and for information what is know about similar problems.
I am not entirely sure whether it will answer your question, but the following references might be of interest to you. First of all, you probably already know about Tao's notes about ergodic theory, in particular Lecture 5: Other topological recurrence results. References [1] and [2] are also well known references. Reference [3] is not about topological semigroups, but it contains many "unavoidability results". See in particular the notion of repetitive semigroups. Actually, I would not be surprised if some results of this book could be proved using ultrafilters.
[1] H. Furstenberg, Y. Katznelson, Idempotents in compact semigroups and Ramsey theory. Israel J. Math. 68 (1989), no. 3, 257--270.
[2] N. Hindman, D. Strauss, Algebra in the Stone-Čech compactification, Theory and applications. Second revised and extended edition, de Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012.
[2] A. de Luca, S. Varricchio, Finiteness and regularity in semigroups and formal languages, Monographs in Theoretical Computer Science. An EATCS Series. Springer-Verlag, Berlin, 1999. x+240 pp. ISBN: 3-540-63771-0