7. Szemerédi's graph regularity lemma II triangle removal lemma
PROFESSOR: I sent out a survey this morning about how the class is going, what you thought of the problem set. And I would appreciate if you provide me some feedback-- so things you like or don't like about the class or about the problem set that was just due last night. So I can try to adjust to make it more interesting and useful for all of you. Last time we talked about Szemerédi's graph regularity lemma. So the regularity lemma, as I mentioned, is an extremely powerful tool in modern combinatorics. And last time we saw the statement and the proof of this regularity lemma. Today, I want to show you how to apply the lemma for extremal applications. In particular, we'll see how to prove Roth's theorem that I mentioned in the very first lecture, about subsets of integers lacking three-term arithmetic progressions. First, let me remind you the regularity lemma. We're always working inside some graph, G. We say that a pair of subsets of vertices is epsi...