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YUFEI ZHAO: We've been spending the past few lectures discussing Szemeredi's Regularity Lemma. And one of the first applications that we discussed of the Regularity Lemma is the triangle removal Lemma. So today, I want to revisit this topic and show you a strengthening of the Removal Lemma for which new regularity techniques are needed. But first, recall the graph removal Lemma. In the graph removal Lemma, we have that for every graph H and epsilon bigger than zero, there exists some delta such that if an N vertex graph has fewer than delta and to the number of vertices of H, many copies of H, then it can be made H-free by removing fewer than epsilon N squared edges. Even in the case when H is a triangle, when this is called a triangle removal Lemma, even in that case, basically the regularity method is more or less the only way that we currently know how to prove this theorem. So we saw this a few lectures ago. What I would like to discuss today is a variant of t...

PROFESSOR: We've been spending quite a few lectures so far discussing Szemeredi's regularity lemma, it's applications in variants of the regularity Lemma. So I want to spend one more lecture, before moving on to a different topic, to tell you about other extensions and other perspectives on Szemeredi's regularity lemma. So hopefully, you all will become experts of the regularity lemma, especially the next homework problem set where there will be plenty of problems for you to practice using the regularity lemma. So one of the things that I would like to discuss today is a hypergraph extension of the triangle removal lemma. So as we saw, the triangle removal lemma was one of the important application of Szemeredi's graph regularity lemma. OK. So that works inside graphs, but now let's go to hypergraphs. In particular, even in the case of three uniform hypergraphs where-- so let's set some terminology. So when r uniform hypergraph, or simply abbre...