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?