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These are asymptotically sharp over an essentially optimal regime for both the uniformity and the number of edges and solve a number of open problems in Extremal Combinatorics. Firstly, we give general conditions under which the crosscut parameter asymptotically determines the Tur\u00e1n number, thus answering a question of Mubayi and Verstra\u00ebte. Secondly, we refine our asymptotic results to obtain several exact results, including proofs of the Huang\u2013Loh\u2013Sudakov conjecture on cross matchings and the F\u00fcredi\u2013Jiang\u2013Seiver conjecture on path expansions. We have introduced two major new tools for the proofs of these results. The first of these, Global Hypercontractivity, is used as a \u2018black box\u2019 (we present it in a separate paper with several other applications). The second tool, presented in this paper, is a far-reaching extension of the Junta Method, which we develop from a powerful and general technique for finding matchings in hypergraphs under certain pseudorandomness conditions.<\/jats:p>","DOI":"10.1007\/s00493-025-00152-4","type":"journal-article","created":{"date-parts":[[2025,4,23]],"date-time":"2025-04-23T13:21:19Z","timestamp":1745414479000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Tur\u00e1n Problems for Expanded Hypergraphs"],"prefix":"10.1007","volume":"45","author":[{"given":"Peter","family":"Keevash","sequence":"first","affiliation":[]},{"given":"Noam","family":"Lifshitz","sequence":"additional","affiliation":[]},{"given":"Eoin","family":"Long","sequence":"additional","affiliation":[]},{"given":"Dor","family":"Minzer","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,4,23]]},"reference":[{"issue":"105","key":"152_CR1","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1093\/qmath\/27.1.25","volume":"27","author":"B\u00e9la Bollob\u00e1s","year":"1976","unstructured":"Bollob\u00e1s, B\u00e9la., Daykin, David E., Erd\u0151s, Paul: Sets of independent edges of a hypergraph. 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