I am a new user here, so forgive me if I am doing anything improperly.

I am currently a masters student in mathematics. I am very interested in combinatorics, and I was thinking about writing my master's thesis about some topic in combinatorics. I have talked to some of the professors at my university, and they are having a slightly difficult time coming up with good research topics for me to look at.

So I am just wondering, are there any accessible (to a masters student) topics in combinatorics/graph theory/etc that you know of?

  • $\begingroup$ Combinatorics covers a lot of territory. What aspects have interested you most? Enumerative combinatorics? Generating functions? Graph theory? (the list goes on...) There are textbooks devoted to each of these and I've only covered part of the domain of combinatorics. Try to identify a domain, and zero in on a more focused problem within that domain. $\endgroup$ – Namaste Apr 23 '13 at 19:48
  • $\begingroup$ My all-time favorite is graph theory. $\endgroup$ – Jamesmiller Apr 23 '13 at 19:48
  • 3
    $\begingroup$ Great! That's a start. For anything you plan to dedicate a lot of time to (like writing a thesis), it's always good when you can zero in on a topic of greatest interest...for motivational purposes, if nothing else! $\endgroup$ – Namaste Apr 23 '13 at 19:51
  • $\begingroup$ That's very good advice! So it's settled. I want to work on graph theory. $\endgroup$ – Jamesmiller Apr 23 '13 at 22:44
  • $\begingroup$ These might be tough, but try openproblemgarden.org/category/combinatorics $\endgroup$ – vadim123 Apr 24 '13 at 4:00

Look for best known combinatorial results, in anything.

The computing power you have access to is a trillion times more powerful that what was available in the 1990's. Also, the algorithms available are much more powerful. For any type of combinatorial problem that needs a computer search, if it hasn't been redone within the last 5 years, it's pretty much is wide open.

I stumbled into this, reading through the Mathematical Games books of Martin Gardner. The Paterson's Worms problem looks interesting -- whoops, we just solved all the open cases. How about the Mrs. Perkins's Quilt problem? I'll just re-use the exact same methods they did in the 1950's. Whoops, vastly extended all known results, and people think I'm an expert.

Dig up neglected problems that were examined with the best cap-guns of their time, and surround them with hydrogen bombs. If a method is given, just reuse it, and see what having 10^20 times more power will do.

But don't rely on computers too much. The Optimal Golomb Ruler project looked at OGR 23 in a distributed project for 4 years, and they rediscovered a result found by a skilled combinatorialist 20 years earlier. Build up skills as both a combinatorialist and as a programmer for best results.

  • $\begingroup$ I really like this answer. +1. $\endgroup$ – Potato Apr 24 '13 at 3:13

Look around for potential advisors with similar interests, get involved in their research. You'll have a very hard time getting an advisor in a topic that isn't more-or-less pressing to them.

My thesis advisor suggested sitting down and writing up short summaries of any relevant papers I found. Writing a 3-5 line summary of a paper means you really understood it. This collection, sans stuff that turned out irrelevant, will be your much prized "state of the art" chapter, and the bibliography will be your starting point.

Learn how to use LaTeX and BibTeX, you will need them. Learn how to use a version control system (I'd suggest git or mercurial), that way you can handle several versions painlessly (and store several copies of the work at some self-hosted or at the school's machines repositories).

Start writing now. It doesn't matter if you are just starting graduate school. What you write today most probably won't survive, but sorting one's ideas by writing them down, looking them over regularly and fixing, throwing out, adding new stuff can be done any time. If the final draft is due Wednesday, sitting in panic in front of the blank screen with nothing else to do does hurt. Been there, gone through it. Still makes me shiver.


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