# Connections between topology and combinatorics

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?

• Just FYI, you have your two links backwards: the first is the combinatorial analogue to Borsuk-Ulam, while the second corresponds to Brouwer. Commented Jun 25, 2021 at 1:15
• Good catch, thanks! Commented Jun 25, 2021 at 1:22

The answer is "yes", and in fact algebraic topology has a lot of combinatorial content. You can find this nowadays with the notion of simplicial sets (and other higher powered tools), but the idea is very old.

In fact, in ye olden days, algebraic topology was called combinatorial topology, with good reason. My personal favorite book on this topic is Henle's A Combinatorial Introduction to Topology. This book proves Brouwer's Fixed Point Theorem, the Jordan Curve Theorem, The Classification Theorem for Compact Surfaces, and many more using combinatorial techniques.

The connection goes both ways, though, and just like we can use combinatorics to solve topological problems, we can use topology to solve combinatorial problems. This observation gives us the field of topological combinatorics, and a great reference for this is Matoušek's Using the Borsuk-Ulam Theorem.

I hope this helps ^_^

• Wonderful! I've ordered a copy of the text, and look forward to working through it. Commented Jun 25, 2021 at 1:25
• Matoušek's book is great. On my shelf it is next to de Longueville's "A Course in Topological Combinatorics", which I mention here because the author is a friend. I would hope that it can be found in a university library. Commented Jun 25, 2021 at 9:29
• dear HallaSurvivor; I have corrected a small typo, I hope that's alright :) (+1) Commented Jul 1, 2021 at 4:36
• @AtticusStonestrom Of course not ^_^. Thanks for the fix! Commented Jul 1, 2021 at 4:39

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:

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

It's quite a fun area of mathematics