TITLE: A General Framework for Ramsey Theory ? 
QUESTION [11 upvotes]: 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?

REPLY [7 votes]: 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."