What is combinatorics? How is it related to Ramsey theory? What is the background needed to study it?

When I was reading about Ramsey theory in some reviews on some books, many people mentioned this branch of mathematics and said that it's related with Ramsey theory and that it's divided into two parts: commutative combinatorics and non-commutative combinatorics.

So, what is combinatorics? How is it related to Ramsey theory? Is it related to groups or mathematical logic? And what is the required background to study it?

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    $\begingroup$ Which parts of the Wikipedia article on combinatorics did you have trouble with? $\endgroup$ – Zev Chonoles Jun 17 '13 at 22:11
  • $\begingroup$ Combinatorics is nice in that it has essentially no prerequisites other than a bit of mathematical maturity (maybe some experience with proofs etc.) and the willingness to look at a lot of problems, just grab a good book and have a go! You will probably find some good recommendations by running a search like this math.stackexchange.com/questions/tagged/…. $\endgroup$ – Alex J Best Jun 17 '13 at 22:21

Check out the Wikipedia entry on Combinatorics. It will include a good distillation of the field(s), since "Combinatorics" is huge and encompasses seemingly unrelated "subfields". The entry will also point to some good references and links. See also Gentle Introduction to Ramsey's Theory , and once you've done that, review Ramsey's Theory, more technical. See also this earlier post: What is Ramsey's Theory? What is its own importance in maths?

As Alex suggests, one of the nice things about combinatorics is that it doesn't require much in the way of prerequisites: you can "dig in" and run with it. You can study sub-topics within Combinatorics that are fairly well-contained, like graph theory, and doing so doesn't require knowledge of every thing that counts as "Combinatorics." See also this earlier post: What is Combinatorics?.

One of my favorite texts in Combinatorics is Peter Cameron's text Combinatorics. But if you search Math.SE using the tags "reference-request" and "Combinatorics", you'll find a host of previously recommended texts.


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