A General Framework for Ramsey Theory ?  There are few results in modern mathematics that I find so deep and full of philosophical implications as Ramsey's theorem. 
I am aware (at some basic level) that it has generated a plethora of further research, going well beyond graph theory, and that there is now an entire industry of Ramsey-like theorems, in many disparate domains (for instance enumerative combinatorics). 
What troubles me, though, is that I do not clearly see what the proper framework for a generalized Ramsey Theory could possibly be. 
If you browse the wiki, you find the following sentence, under the voice RAMSEY THEORY:
Problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?"
That sound general enough, but a little too informal: 
can this sentence be re-formulated in a rigorous way? 
I mean: categories of structured sets are the bread-and-butter of category theory (think for instance of categories of algebras, categories of ordered sets, etc), so it looks to me as if there could be a convenient formulation of the  quoted sentence in suitable categorical form 
(something like: if .... then for every object of the category there is a large -in some suitable sense- sub-object satisfying ....., fill the dots)
Anything out there? 
 A: You might be interested in the 2010 book "Introduction to Ramsey Spaces" by Stevo Todorcevic
"Ramsey theory is a fast-growing area of combinatorics with deep connections to other fields of mathematics such as topological dynamics, ergodic theory, mathematical logic, and algebra. The area of Ramsey theory dealing with Ramsey-type phenomena in higher dimensions is particularly useful. Introduction to Ramsey Spaces presents in a systematic way a method for building higher-dimensional Ramsey spaces from basic one-dimensional principles. It is the first book-length treatment of this area of Ramsey theory, and emphasizes applications for related and surrounding fields of mathematics, such as set theory, combinatorics, real and functional analysis, and topology. In order to facilitate accessibility, the book gives the method in its axiomatic form with examples that cover many important parts of Ramsey theory both finite and infinite."
A: "Complete disorder is impossible" T. Motzkin
Every large system contains a large well organized subsystem.
"How large" can be a great question (assuming we take it to mean large finite) but just showing that a certain class of systems has this property seems deeper to me. I find the audacity of the induction involved stunning. Sometimes the cleanest path to finite results is to replace "large" by "infinite", prove that result, and then concluded that the finite results hold. Some results are deep, in part, because we talk about sizes in ranges we need novel notation to even mention. 
A: I do not really understand the motivation behind the question. Ramsey's Theorem is just a very beautiful piece of mathematics, and there is no need to generalize it in terms of category theory just for the sake of generalization. On the other hand, the note Ramsey Theory by R. L. Graham and B. L. Rothschild discusses categorical generalizations.
Let $C$ be a category (which should be well-powered). For objects $N,K \in C$ denote by $\binom{N}{K}$ set of subobjects of $N$ of the form $K \to N$. Every monomorphism $\phi : L \to N$ induces a map $\tilde{\phi} : \binom{L}{K} \to \binom{N}{K}$. The category $C$ is called Ramsey if for every positive integer $r$ ("number of colors") and all objects $K,L \in C$ there is an object $N \in C$ ("Ramsey number") such that for every map $c : \binom{N}{K} \to [1,r]$ ("coloring") there exists a monomorphism $\phi : L \to N$ and some $i \in [1,r]$ such that the diagram
$$\begin{matrix} \binom{N}{K} & \xrightarrow{c} & [1,r] \\\\ {\small \tilde{\phi}} \uparrow ~ & & \uparrow \\\\ \binom{L}{K} & \xrightarrow{ } & \{i\} \end{matrix}$$
commutes ("there is a complete monochromatic subgraph on $L$ vertices").
For the category of finite sets and with injections as morphisms this is precisely Ramsey's classical Theorem (the case $K=2$ being about graphs). For the category of finite-dimensional vector spaces over a finite field one discovers Rota's conjecture, which is proven over $\mathbb{F}_2,\mathbb{F}_3,\mathbb{F}_4$, but otherwise unknown. This already indicates how hard it will be to test the Ramsey property of other categories.
See also the paper Ramsey's Theorem for a class of categories by Graham, Leeb, Rotschild.
