{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T18:57:51Z","timestamp":1757617071517,"version":"3.44.0"},"reference-count":6,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T00:00:00Z","timestamp":1753228800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T00:00:00Z","timestamp":1753228800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2025,8]]},"abstract":"Abstract<\/jats:title>\n We prove that given a constant \n $$k \\ge 2$$<\/jats:tex-math>\n <\/jats:inline-formula> and a large set system \n $$\\mathcal {F}$$<\/jats:tex-math>\n <\/jats:inline-formula> of sets of size at most w<\/jats:italic>, a typical k<\/jats:italic>-tuple of sets \n $$(S_1, \\cdots, S_k)$$<\/jats:tex-math>\n <\/jats:inline-formula> from \n $$\\mathcal {F}$$<\/jats:tex-math>\n <\/jats:inline-formula> can be \u201cblown up\u201d in the following sense: for each \n $$1 \\le i \\le k$$<\/jats:tex-math>\n <\/jats:inline-formula>, we can find a large subfamily \n $$\\mathcal {F}_i$$<\/jats:tex-math>\n <\/jats:inline-formula> containing \n $$S_i$$<\/jats:tex-math>\n <\/jats:inline-formula> so that for \n $$i \\ne j$$<\/jats:tex-math>\n <\/jats:inline-formula>, if \n $$T_i \\in \\mathcal {F}_i$$<\/jats:tex-math>\n <\/jats:inline-formula> and \n $$T_j \\in \\mathcal {F}_j$$<\/jats:tex-math>\n <\/jats:inline-formula>, then \n $$T_i \\cap T_j=S_i \\cap S_j$$<\/jats:tex-math>\n <\/jats:inline-formula>. We also show that the answer to the multicolor version of the sunflower conjecture is the same as the answer for the original, up to an exponential factor.<\/jats:p>","DOI":"10.1007\/s00493-025-00163-1","type":"journal-article","created":{"date-parts":[[2025,7,23]],"date-time":"2025-07-23T14:18:54Z","timestamp":1753280334000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Set System Blowups"],"prefix":"10.1007","volume":"45","author":[{"given":"Ryan","family":"Alweiss","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,23]]},"reference":[{"issue":"3","key":"163_CR1","doi-asserted-by":"publisher","first-page":"795","DOI":"10.4007\/annals.2021.194.3.5","volume":"194","author":"R Alweiss","year":"2021","unstructured":"Alweiss, R., Lovett, S., Wu, K., Zhang, J.: Improved bounds for the sunflower lemma. Ann. of Math. 194(3), 795\u2013815 (2021)","journal-title":"Ann. of Math."},{"issue":"3","key":"163_CR2","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/BF02579328","volume":"1","author":"M Deza","year":"1981","unstructured":"Deza, M., Frankl, P.: Every large set of equidistant (0, +1, -1)-vectors forms a sunflower. Combinatorica 1(3), 225\u2013231 (1981)","journal-title":"Combinatorica"},{"key":"163_CR3","doi-asserted-by":"crossref","unstructured":"Frankston, K., Kahn, J., Narayanan, B., Park, J.: Thresholds versus fractional expectation-thresholds, Ann. of Math. (2) 194(2), 475\u2013495 (2021)","DOI":"10.4007\/annals.2021.194.2.2"},{"issue":"1","key":"163_CR4","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1112\/jlms\/s1-35.1.85","volume":"35","author":"P Erd\u0151s","year":"1960","unstructured":"Erd\u0151s, P., Rado, E.: Intersection theorems for systems of sets. J. Lond. Math. Soc. 35(1), 85\u201390 (1960)","journal-title":"J. Lond. Math. Soc."},{"key":"163_CR5","doi-asserted-by":"crossref","unstructured":"Kleinberg, R., Sawin, W.F., Speyer, D.E.: The growth rate of tri-colored sum-free sets. Discret. Anal. 12, 10 (2018)","DOI":"10.19086\/da.3734"},{"key":"163_CR6","unstructured":"Rao, A.: Coding for Sunflowers. Discret. Anal. (2020)"}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-025-00163-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00493-025-00163-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-025-00163-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T00:06:00Z","timestamp":1757117160000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00493-025-00163-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,23]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["163"],"URL":"https:\/\/doi.org\/10.1007\/s00493-025-00163-1","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"type":"print","value":"0209-9683"},{"type":"electronic","value":"1439-6912"}],"subject":[],"published":{"date-parts":[[2025,7,23]]},"assertion":[{"value":"17 August 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"8 May 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 July 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"41"}}