{"id":6229,"date":"2026-06-25T14:34:55","date_gmt":"2026-06-25T05:34:55","guid":{"rendered":"https:\/\/www.ibs.re.kr\/ecopro\/?page_id=6229"},"modified":"2026-07-31T13:14:24","modified_gmt":"2026-07-31T04:14:24","slug":"summer-2026","status":"publish","type":"page","link":"https:\/\/www.ibs.re.kr\/ecopro\/summer-2026\/","title":{"rendered":"Summer 2026"},"content":{"rendered":"\n<ul id=\"block-5ee26a2a-7c74-446e-be1a-e5066c1642b7\" 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>IBS ECOPRO <strong>B332<\/strong> (Main building 3rd floor).<\/em><\/li>\n\n\n\n<li><strong>Jul 6 &#8211; <strong>Jul <\/strong>10<\/strong>, Entropy method in extremal combinatorics<\/li>\n\n\n\n<li><strong><strong>Jul 27 &#8211; <strong>Jul <\/strong>31<\/strong><\/strong>, Hypergraphs with Bounded VC-Dimension: Incidences, Zarankiewicz Problems, and Radon-Type Theorems<\/li>\n\n\n\n<li><strong>Zoom: 3469344087, PW: 202607<\/strong><\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><br>Entropy method in extremal combinatorics<\/h2>\n\n\n\n<h5 class=\"wp-block-heading\">Ting-Wei Chao,  Massachusetts Institute of Technology<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\"><em>The entropy method has been used in many recent works in extremalcombinatorics. With the help of Shannon entropy, significant progresshas been made on several classical problems, such as the union-closed conjecture and Sidorenko&#8217;s conjecture. There are also recent new proofs of classic theorems and their generalizations using entropy, such as the Kruskal-Katona theorem and Turan&#8217;s theorem. We will cover the basics of entropy and talk about some applications to graph homomorphism counting problems.<\/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-f6132678-57ed-4183-8fca-bd36f73f1431\"><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\/BV1kZTS6wEnW\/\" 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\/6 Lecture 1: Basics, bitwise information, and union-closed conjecture <strong><a type=\"link\" href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/01.pdf\" id=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/01.pdf\"><span style=\"color: #fe6a00;\" class=\"stk-highlight\">NOTE 1<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/7 Lecture 2: Graph homomorphism inequalities and Sidorenko&#8217;s conjecture &nbsp;<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/02.pdf\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 2<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/8 Lecture 3: Shearer&#8217;s inequality and its applications &nbsp;<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/03.pdf\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 3<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/9 Lecture 4: Friedgut&#8211;Kahn theorem and mixture bound (I) <strong><strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/04.pdf\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 4<\/span><\/a><\/strong><\/strong><\/li>\n\n\n\n<li>7\/10 Lecture 5: Mixture bound (II)&nbsp;<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/05.pdf\"><span style=\"color: #ff6a00;\" 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\"><br>Hypergraphs with Bounded VC-Dimension: Incidences, Zarankiewicz Problems, and Radon-Type Theorems<\/h2>\n\n\n\n<h5 class=\"wp-block-heading\">Shakhar Smorodinsky, Ben-Gurion University<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\"><em>The notion of VC-dimension, introduced in statistical learning theory, has become a central tool inmodern combinatorics, discrete geometry, and theoretical computer science. It provides a powerful way to measure the complexity of set systems, or equivalently hypergraphs, and many natural geometric hypergraphs have bounded VC-dimension. In this course, I will discuss the rich combinatorial theory of hypergraphs with bounded VC-dimension and some of its geometric manifestations. We will begin with the Sauer-Shelah-Perles lemma and its basic consequences and then move to epsilon-nets and related sampling and packing phenomena. We will studyextremal questions for graphs and hypergraphs of bounded VC-dimension, including Zarankiewicz-type problems and incidence bounds, together with tools such as the crossing lemma and polynomial partitioning. Further topics will include matchings with low crossing numbers, generalizations of epsilon-nets, and Radon- and Tverberg-type theorems for unions of convex sets.<\/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-6509b65f-8969-4a32-af38-28d332d37d37\"><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\/BV1CegZ6hEHK\/\" 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\/27 Lecture 1: Basic definitions: VC-dimension: examples arising in geometry and combinatorics. <strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/main-lecture-1.pptx\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 1<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/28 Lecture 2: The shatter function, Sauer-Shelah-Perles Lemma. applications to boolean enrichment of hypergraphs. \u00a0<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/main-lecture-2-1.pptx\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 2<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/29 Lecture 3:  Hypergraphs defined by polynomials of bounded degree, Basic convexity and the relation between Radon&#8217;s Theorem and VC-dimension of halfspaces, abstract convexity spaces. \u00a0<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/main-lecture-3.pptx\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 3<\/span><\/a><\/strong><\/li>\n\n\n\n<li>7\/30 Lecture 4: Application of the VC-dimension lens to union of convex sets: Extending Radon and Tverberg theorems to unions of convex sets. <strong><strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/main-lecture-4.pptx\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 4<\/span><\/a><\/strong><\/strong><\/li>\n\n\n\n<li>7\/31 Lecture 5: Epsilon nets and an extension to epsilon-t-nets. Applications to Zarankiewicz problem for bounded VC-dimension graphs. \u00a0<strong><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/07\/main-lecture-5.pptx\"><span style=\"color: #ff6a00;\" class=\"stk-highlight\">NOTE 5<\/span><\/a><\/strong><\/li>\n<\/ul>\n\n\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"723\" height=\"1024\" src=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3-723x1024.png\" alt=\"\" class=\"wp-image-6257\" srcset=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3-723x1024.png 723w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3-212x300.png 212w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3-768x1087.png 768w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3-283x400.png 283w, https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3.png 1054w\" sizes=\"auto, (max-width: 723px) 100vw, 723px\" \/><\/figure>\n","protected":false},"excerpt":{"rendered":"<p>Entropy method in extremal combinatorics Ting-Wei Chao, Massachusetts Institute of Technology The entropy method has been used in many recent works in extremalcombinatorics. With the help of Shannon entropy, significant &hellip; <\/p>\n<p class=\"link-more\"><a href=\"https:\/\/www.ibs.re.kr\/ecopro\/summer-2026\/\" class=\"more-link\">Continue reading<span class=\"screen-reader-text\"> &#8220;Summer 2026&#8221;<\/span><\/a><\/p>\n","protected":false},"author":5,"featured_media":6257,"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-6229","page","type-page","status-publish","has-post-thumbnail","hentry"],"featured_image_src":"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3.png","yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Summer 2026 - 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-2026\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Summer 2026 - Extremal Combinatorics and Probability Group\" \/>\n<meta property=\"og:description\" content=\"Entropy method in extremal combinatorics Ting-Wei Chao, Massachusetts Institute of Technology The entropy method has been used in many recent works in extremalcombinatorics. With the help of Shannon entropy, significant &hellip; Continue reading &quot;Summer 2026&quot;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.ibs.re.kr\/ecopro\/summer-2026\/\" \/>\n<meta property=\"og:site_name\" content=\"Extremal Combinatorics and Probability Group\" \/>\n<meta property=\"article:modified_time\" content=\"2026-07-31T04:14:24+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.ibs.re.kr\/ecopro\/wp-content\/uploads\/2026\/06\/6a6c177162f9a8da3bee9779de6a8dd2-3.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1054\" \/>\n\t<meta property=\"og:image:height\" content=\"1492\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"3 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/\",\"url\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/\",\"name\":\"Summer 2026 - Extremal Combinatorics and Probability Group\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/wp-content\\\/uploads\\\/2026\\\/06\\\/6a6c177162f9a8da3bee9779de6a8dd2-3.png\",\"datePublished\":\"2026-06-25T05:34:55+00:00\",\"dateModified\":\"2026-07-31T04:14:24+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/#primaryimage\",\"url\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/wp-content\\\/uploads\\\/2026\\\/06\\\/6a6c177162f9a8da3bee9779de6a8dd2-3.png\",\"contentUrl\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/wp-content\\\/uploads\\\/2026\\\/06\\\/6a6c177162f9a8da3bee9779de6a8dd2-3.png\",\"width\":1054,\"height\":1492},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/summer-2026\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Summer 2026\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/#website\",\"url\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/\",\"name\":\"Extremal Combinatorics and Probability Group\",\"description\":\"\uae30\ucd08\uacfc\ud559\uc5f0\uad6c\uc6d0 \uadf9\ub2e8 \uc870\ud569 \ubc0f \ud655\ub960 \uadf8\ub8f9\",\"publisher\":{\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/#organization\",\"name\":\"IBS Extremal Combinatorics and Probability Group\",\"url\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/wp-content\\\/uploads\\\/2022\\\/01\\\/ibs-ecopro-logo.png\",\"contentUrl\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/wp-content\\\/uploads\\\/2022\\\/01\\\/ibs-ecopro-logo.png\",\"width\":784,\"height\":197,\"caption\":\"IBS Extremal Combinatorics and Probability Group\"},\"image\":{\"@id\":\"https:\\\/\\\/www.ibs.re.kr\\\/ecopro\\\/#\\\/schema\\\/logo\\\/image\\\/\"}}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Summer 2026 - Extremal Combinatorics and Probability Group","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.ibs.re.kr\/ecopro\/summer-2026\/","og_locale":"en_US","og_type":"article","og_title":"Summer 2026 - Extremal Combinatorics and Probability Group","og_description":"Entropy method in extremal combinatorics Ting-Wei Chao, Massachusetts Institute of Technology The entropy method has been used in many recent works in extremalcombinatorics. 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