TITLE: Connections between topology and combinatorics
QUESTION [10 upvotes]: I recently learned that both Brouwer's fixed point theorem and Borsuk-Ulam admit fairly simple graph-theoretic proofs.
While I am able to follow the gist of both proofs, I find myself yearning for some kind of larger context to fit them in.
Is there any sort of general machinery for converting topological problems (or, at least, some class of topological problems) into combinatorial ones that these follow from?

REPLY [2 votes]: One of the other areas where Topology crosses with Combinatorics is the area of Topological Dynamical Systems, where one can prove Ramsey Type theorems using topological tools.
Theorems that can be proven include:

Van der Waerdens theorem about prime subsequences
Infinite variant of the Ramsey Theorem
Hindman's theorem.

It's quite a fun area of mathematics