Perfect packings in quasirandom hypergraphs I

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• 作者：John Lenz¹ ; ^{lenz@math.uic.edu" class="auth_mail" title="E-mail the corresponding author} ; Dhruv Mubayi² ; ^{mubayi@math.uic.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Quasirandom hypergraph ; Hypergraph packing ; Linear hypergraph ; Hypergraph matching ; Absorbing method
• 刊名：Journal of Combinatorial Theory, Series B
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：119
• 期：Complete
• 页码：155-177
• 全文大小：529 K

文摘

Let k≥2$k\ge 2$ and F be a linear k-uniform hypergraph with v vertices. We prove that if n   is sufficiently large and v|n$v|n$, then every quasirandom k-uniform hypergraph on n   vertices with constant edge density and minimum degree Ω(n^k−1)$\mathrm{\Omega }(n^{k-1})$ admits a perfect F  -packing. The case k=2$k=2$ follows immediately from the blowup lemma of Komlós, Sárközy, and Szemerédi. We also prove positive results for some nonlinear F but at the same time give counterexamples for rather simple F that are close to being linear. Finally, we address the case when the density tends to zero, and prove (in analogy with the graph case) that sparse quasirandom 3-uniform hypergraphs admit a perfect matching as long as their second largest eigenvalue is sufficiently smaller than the largest eigenvalue.