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IBS ECOPRO 2025 Summer Courses: Frontiers of Combinatorics https://www.ibs.re.kr/ecopro/summer-2... Lior Gishboliner, Around Removal Lemmas (Lecture 1) July 28, 2025, @ 10:00AM - 11:30AM KST IBS room B332 SPEAKER Lior Gishboliner University of Toronto, Canada https://sites.google.com/view/lior-gi... The removal lemma for a graph (or hypergraph) property P states that if a graph 𝐺 is 𝜀-far from satisfying P, then a random sample of 𝑓(𝜀) vertices of 𝐺 is likely to not satisfy P. Such results are useful both in combinatorics and theoretical computer science, where they correspond to property testing algorithms. Removal lemmas are typically proved using a regularity lemma, which leads to very poor bounds on 𝑓(𝜀). A central problem is to understand for which properties P we can take 𝑓(𝜀) to be polynomial in 𝜀. The goal of this course is to give an overview of regularity and removal lemmas and present the tools used in the study of this question, including some recent results.