More Short Stories
Sunflower Lemma
The sunflower lemma is one of those wonderful theorems that tells you that in a "sufficiently large structure there must live some very special structure." The inductive proof is simple and yet characteristic of arguments in external combinatorics. With the right induction hypothesis, the special case puts the general results dead in its sights.
Concentration Inequalities
Concentration inequalities are the heart and soul of many probability arguments, especially those that engage probability and combinatorics. Svante Janson is one of the grandmaster's of this business and he has written a very nice, friendly, and brief introduction to the "what everybody needs to know."
Erdos-Newman Disc Exhaustion Theorem
The question asked by Erdos and Newman is this: How fasts can one empty out a square if you remove one disk at a time? The paper uses only elementary geometry and calculus, but it is devilishly clever. Putting it into a "Problem Solution" format would be very instructive and potentially useful.
The Erdos Newman method is both deterministic and greedy. Can you beat them by using randomization? Can you beat them by being non-greedy?
(Caveat: This is a good learning project, but perhaps a less good research project unless you just fall in love with it. If you do fall in love, be sure to do a literature search before spending a ton of effort. Incidentally, this is always good advice.)
Kannan's Improved Concentration Inequalties
The Azuma-Hoffding inequality is one of the workhorses of probability in combinatorics. Ravi Kannan has an inequality that does better in some important cases. It depends on a somewhat delicate induction and dynamic program. It would be a great thing to (a) understand (b) simplify or (c) apply. Knowing such inequaliies is on of the themes of the LMG.