SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1813-1829, January 2020.
In many proofs concerning extremal parameters of Berge hypergraphs one starts with analyzing that part of that shadow graph which is contained in many hyperedges. Capturing this phenomenon we introduce two new types of hypergraphs. A hypergraph ${\mathcal H}$ is a $t$-heavy copy of a graph $F$ if there is a copy of $F$ on its vertex set such that each edge of $F$ is contained in at least $t$ hyperedges of ${\mathcal H}$. ${\mathcal H}$ is a $t$-wise Berge copy of $F$ if additionally for distinct edges of $F$ those $t$ hyperedges are distinct. We extend known upper bounds on the Turán number of Berge hypergraphs to the $t$-wise Berge hypergraphs case. We asymptotically determine the Turán number of $t$-heavy and $t$-wise Berge copies of long paths and cycles and exactly determine the Turán number of $t$-heavy and $t$-wise Berge copies of cliques. In the case of 3-uniform hypergraphs, we consider the problem in more details and obtain additional results.

中文翻译：

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$ t $-明智的贝尔和$ t $-超重图

SIAM离散数学杂志，第34卷，第3期，第1813-1829页，2020年1月。
在许多有关Berge超图的极值参数的证明中，首先要分析阴影图中包含在许多超边中的那部分。为捕捉这种现象，我们引入了两种新的超图。超图$ {\ mathcal H} $是图$ F $的$ t $重型副本，如果在其顶点集上有$ F $的副本，使得$ F $的每个边都包含在至少$ $ {\ mathcal H} $的t $超边。$ {\ mathcal H} $是$ F $的$ t $明智的Berge副本，如果另外为$ F $的不同边而那些$ t $超边是不同的。我们将Berge超图的Turán数的已知上限扩展到$ t $ -wise Berge超图的情况。我们渐近地确定长路径和周期的$ t-重型和$ t-明智Berge副本的Turán数，并精确确定集团的$ t $-重型和$ t $-明智Berge副本的Turán数。