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?
Best Answer
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.