Asymptotics for Turan numbers of cycles in 3-uniform linear hypergraphs

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:163

Article

Ergemlidze, Beka¹  Gyori, Ervin^1,2  Methuku, Abhishek¹ 

[1] Cent European Univ, Dept Math, Budapest, Hungary

[2] Hungarian Acad Sci, Renyi Inst, Budapest, Hungary

关键词: Linear Turan number;    Berge cycle;    Five cycle;    Linear cycle;    Loose cycle;    Hypergraph Turan problem;   

DOI  :  10.1016/j.jcta.2018.12.004

来源: Elsevier

【 摘 要 】

Let F be a family of 3-uniform linear hypergraphs. The linear Turan number of F is the maximum possible number of edges in a 3-uniform linear hypergraph on n vertices which contains no member of F as a subhypergraph. In this paper we show that the linear Turan number of the five cycle C-5 (in the Berge sense) is 1/3 root 3n(3/2) asymptotically. We also show that the linear Turan number of the four cycle C-4 and {C-3,C-4} are equal asymptotically, which is a strengthening of a theorem of Lazebnik and Verstraete [16]. We establish a connection between the linear Turan number of the linear cycle of length 2k + 1 and the extremal number of edges in a graph of girth more than 2k - 2. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang [8], we obtain that the linear Turan number of the linear cycle of length 2k + 1 is circle minus(n(1+1/k)) for k = 2, 3, 4, 6. (C) 2018 Elsevier Inc. All rights reserved.

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