{"id":5317,"date":"2025-05-16T09:17:06","date_gmt":"2025-05-16T00:17:06","guid":{"rendered":"https:\/\/www.ibs.re.kr\/ecopro\/?page_id=5317"},"modified":"2025-08-09T16:12:48","modified_gmt":"2025-08-09T07:12:48","slug":"summer-2025","status":"publish","type":"page","link":"https:\/\/www.ibs.re.kr\/ecopro\/summer-2025\/","title":{"rendered":"Summer 2025"},"content":{"rendered":"\n<ul class=\"wp-block-list\">\n<li>Time: 9:00-10:30am (Beijing), 10:00-11:30am(Seoul)<\/li>\n\n\n\n<li>Place: <em><span style=\"color: #8e1e1e;\" class=\"stk-highlight\">IBS ECOPRO <strong>B332<\/strong> (Main building 3rd floor).<\/span><\/em><\/li>\n\n\n\n<li><strong>Jul 28 &#8211; Aug 1<\/strong>, Around removal lemmas<\/li>\n\n\n\n<li><strong>Aug 4 &#8211; 8<\/strong>, Introduction to Boolean analysis<\/li>\n\n\n\n<li><strong><span style=\"color: #0693e3;\" class=\"stk-highlight\">Zoom: 3469344087, PW: 2025<\/span><\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\"><br>Around removal lemmas<\/h2>\n\n\n\n<h5 class=\"wp-block-heading\">Lior Gishboliner, University of Toronto, Canada<\/h5>\n\n\n\n<p><em>The removal lemma for a graph (or hypergraph) property $\\mathcal{P}$ states that if a graph $G$ is $\\varepsilon$-far from satisfying $\\mathcal{P}$, then a random sample of $f(\\varepsilon)$ vertices of $G$ is likely to not satisfy $\\mathcal{P}$. 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 $f(\\varepsilon)$. A central problem is to understand for which properties $\\mathcal{P}$ we can take $f(\\varepsilon)$ to be polynomial in $\\varepsilon$. 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.<\/em><\/p>\n\n\n<div style=\"gap: 20px;\" class=\"align-button-center ub-buttons orientation-button-row ub-flex-wrap wp-block-ub-button\" id=\"ub-button-49d223c2-dc2d-41b4-b1c2-311952cb619b\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/08\/IBS_2025Summer_removal_lemma.pdf\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #cf2e2e; --ub-button-border: 3px solid #cf2e2e; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; font-size: 17px; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-top-left-radius: 100px;; border-top-right-radius: 100px;; border-bottom-left-radius: 100px;; border-bottom-right-radius: 100px;; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Note<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.bilibili.com\/video\/BV1Hz87zyEQ6\/\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #0693e3; --ub-button-border: 3px solid #0693e3; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; font-size: 17px; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-top-left-radius: 20%;; border-top-right-radius: 20%;; border-bottom-left-radius: 20%;; border-bottom-right-radius: 20%;; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Videos<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>7\/28 Lecture 1: Szemeredi&#8217;s regularity lemma and the counting lemma, proof of the triangle removal lemma. <a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/07\/IBS_Summer_Course-July-28.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\"><strong>NOTE 1<\/strong><\/span><\/a><\/li>\n\n\n\n<li>7\/29 Lecture 2: Behrend&#8217;s construction, the $H$-removal lemma is polynomial if and only if $H$ is bipartite (Alon&#8217;s theorem).&nbsp;<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/07\/IBS_Summer_Course-July-29.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">NOTE 2<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/30 Lecture 3: The induced removal lemma. VC-dimension, the Sauer-Shelah lemma.&nbsp;<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/08\/IBS_Summer_Course-July-30-31.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">NOTE 3-4<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/31 Lecture 4: Ultra-strong regularity for graphs of bounded VC-dimension, the infinite removal lemma, property testing.<\/li>\n\n\n\n<li>8\/1 Lecture 5: Testing bipartiteness, testing for large independent sets, hypergraph regularity and VC-dimension..&nbsp;<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/08\/IBS_Summer_Course-August-1.pdf\"><span style=\"color: #ff6900;\" class=\"stk-highlight\">NOTE 5<\/span><\/a><\/strong><\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<h2 class=\"wp-block-heading\">Introduction to Boolean analysis<\/h2>\n\n\n\n<h5 class=\"wp-block-heading\">Yuval Filmus, Technion, Israel<\/h5>\n\n\n\n<p><em>Boolean function analysis studies Boolean functions on the Boolean cube and other domains from a spectral perspective.<\/em> <em>Motivated initially by social choice theory, it has seen many applications in combinatorics and theoretical computer science.<\/em> <em>We will cover the basic definitions and results, and see some classical applications as well as some more recent ones.<\/em><\/p>\n\n\n<div style=\"gap: 20px;\" class=\"align-button-center ub-buttons orientation-button-row ub-flex-wrap wp-block-ub-button\" id=\"ub-button-d3c348bb-da07-4605-8a4a-1c9beb45735f\"><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/08\/Introduction_to_Boolean_Analysis__ECOPRO_-1.pdf\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #cf2e2e; --ub-button-border: 3px solid #cf2e2e; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-top-left-radius: 100px;; border-top-right-radius: 100px;; border-bottom-left-radius: 100px;; border-bottom-right-radius: 100px;; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Note<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><div class=\"ub-button-container\">\n\t\t\t<a href=\"https:\/\/www.bilibili.com\/video\/BV1dBtMziEPZ\/\" target=\"_blank\" rel=\"noopener noreferrer nofollow\" class=\"ub-button-block-main   ub-button-flex\" role=\"button\" style=\"--ub-button-background-color: transparent; --ub-button-color: #0693e3; --ub-button-border: 3px solid #0693e3; --ub-button-hover-color: #313131; --ub-button-hover-border: 3px solid #313131; font-size: 17px; padding-top: 10px; padding-right: 10px; padding-bottom: 10px; padding-left: 10px; border-top-left-radius: 20%;; border-top-right-radius: 20%;; border-bottom-left-radius: 20%;; border-bottom-right-radius: 20%;; \">\n\t\t\t\t<div class=\"ub-button-content-holder\" style=\"flex-direction: row\">\n\t\t\t\t\t<span class=\"ub-button-block-btn\"><strong>Videos<\/strong><\/span>\n\t\t\t\t<\/div>\n\t\t\t<\/a>\n\t\t<\/div><\/div>\n\n\n<ul class=\"wp-block-list\">\n<li>8\/4 Lecture 1: Introduction and linearity testing<\/li>\n\n\n\n<li>8\/5 Lecture 2: KKL theorem<\/li>\n\n\n\n<li>8\/6 Lecture 3: Hypercontractivity<\/li>\n\n\n\n<li>8\/7 Lecture 4: Biased Fourier analysis<\/li>\n\n\n\n<li>8\/8 Lecture 5: Global hypercontractivity, Bourgain\u2019s booster theorem and Erd\u0151s-Ko-Rado<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<div class=\"wp-block-stackable-image stk-block-image stk-block stk-acbc9cc\" data-block-id=\"acbc9cc\"><figure><span class=\"stk-img-wrapper stk-image--shape-stretch\"><img loading=\"lazy\" decoding=\"async\" class=\"stk-img wp-image-5327\" src=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/summer-school-2025.jpg\" width=\"1280\" height=\"1811\" srcset=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/summer-school-2025.jpg 1280w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/summer-school-2025-212x300.jpg 212w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/summer-school-2025-724x1024.jpg 724w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/summer-school-2025-768x1087.jpg 768w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/summer-school-2025-1086x1536.jpg 1086w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/summer-school-2025-283x400.jpg 283w\" sizes=\"auto, (max-width: 767px) 89vw, (max-width: 1000px) 54vw, (max-width: 1071px) 543px, 580px\" \/><\/span><\/figure><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Around removal lemmas Lior Gishboliner, University of Toronto, Canada The removal lemma for a graph (or hypergraph) property $\\mathcal{P}$ states that if a graph $G$ is $\\varepsilon$-far from satisfying $\\mathcal{P}$, &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/summer-2025\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Summer 2025&#8221;<\/span><\/a><\/p>\n","protected":false},"author":5,"featured_media":5332,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_uag_custom_page_level_css":"","_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"class_list":["post-5317","page","type-page","status-publish","has-post-thumbnail","hentry"],"featured_image_src":"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2025\/05\/2025summer-2.png","yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.6 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Summer 2025 - Extremal Combinatorics and Probability Group<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.ibs.re.kr\/ecopro\/summer-2025\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Summer 2025 - Extremal Combinatorics and Probability Group\" \/>\n<meta property=\"og:description\" content=\"Around removal lemmas Lior Gishboliner, University of Toronto, Canada The removal lemma for a graph (or hypergraph) property $mathcal{P}$ states that if a graph $G$ is $varepsilon$-far from satisfying $mathcal{P}$, &hellip; 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