{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,12,12]],"date-time":"2024-12-12T05:58:07Z","timestamp":1733983087113,"version":"3.30.2"},"reference-count":14,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2003,2,28]],"date-time":"2003-02-28T00:00:00Z","timestamp":1046390400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Random Struct Algorithms"],"published-print":{"date-parts":[[2003,5]]},"abstract":"Abstract<\/jats:title>Let \ud835\udca50<\/jats:sub>be any fixed 3\u2010uniform hypergraph. For a 3\u2010uniform hypergraph \u210b\ufe01 we define \u03bd(\u210b\ufe01) to be the maximum size of a set of pairwise triple\u2010disjoint copies of \ud835\udca50<\/jats:sub>in \u210b\ufe01. We say a function \u03c8 from the set of copies of \ud835\udca50<\/jats:sub>in \u210b\ufe01 to [0, 1] is afractional<\/jats:italic>\ud835\udca50<\/jats:sub>\u2010packing<\/jats:italic>of \u210b\ufe01 if \u2211\ud835\udca5\u220be<\/jats:italic><\/jats:sub>\u03c8(\ud835\udca5) \u2264 1 for every triplee<\/jats:italic>of \u210b\ufe01. Then \u03bd(\u210b\ufe01) is defined to be the maximum value of \u2211\u03c8(\ud835\udca5) over all fractional \ud835\udca50<\/jats:sub>\u2010packings \u03c8 of \u210b\ufe01. We show that \u03bd(\u210b\ufe01) \u2212 \u03bd(\u210b\ufe01) =o<\/jats:italic>(|V<\/jats:italic>(\u210b\ufe01)|3<\/jats:sup>) for all 3\u2010uniform hypergraphs \u210b\ufe01. This extends the analogous result for graphs, proved by Haxell and R\u00f6dl (2001), and requires a significant amount of new theory about regularity of 3\u2010uniform hypergraphs. In particular, we prove a result that we call the Extension Theorem. This states that if ak<\/jats:italic>\u2010partite 3\u2010uniform hypergraph is regular [in the sense of the hypergraph regularity lemma of Frankl and R\u00f6dl (2002)], then almost every triple is in about the same number of copies ofK<\/jats:italic>(the complete 3\u2010uniform hypergraph withk<\/jats:italic>vertices). \u00a9 2003 Wiley Periodicals, Inc. Random Struct. 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