SIAM Journal on Discrete Mathematics, Volume 34, Issue 3, Page 1485-1492, January 2020.
For a graph $G=(V,E)$, a hypergraph $H$ is called a Berge-$G$ if there is a bijection $f:E(G)\mapsto E(H)$ such that $e\subseteq f(e)$ for all $e\in E(G)$. The family of Berge-$G$ hypergraphs is denoted by $\mathcal{B}(G)$. The maximum number of edges in an $n$-vertex $r$-graph with no subhypergraph isomorphic to any Berge-$G$ is denoted by $ex_r(n, \mathcal{B}(G))$. Gyárfás [SIAM J. Discrete Math., 33 (2019), pp. 383--392] showed that for $n\geq 6$, $ex_3(n,\mathcal{B}(K_4))=\lfloor\frac{n}{3}\rfloor\lfloor\frac{n+1}{3}\rfloor\lfloor\frac{n+2}{3}\rfloor$. However, we found an error in the proof of the result when $n\ge 7$. A recent result due to Gerbner, Methuku, and Palmer [European J. Combin., 86 (2020), 103082] implies that for $n\geq 9$, $ex_3(n,\mathcal{B}(K_4))=\lfloor\frac{n}{3}\rfloor\lfloor\frac{n+1}{3}\rfloor\lfloor\frac{n+2}{3}\rfloor$. In this paper we prove the remaining cases $n=7$ and $n=8$ for the completeness of the conclusion.

中文翻译：

---

3均匀超图中的TuránBerge- $ K_4 $

SIAM离散数学杂志，第34卷，第3期，第1485-1492页，2020年1月。
对于图$ G =（V，E）$，如果存在双射$ f：E（G）\ mapsto E（H）$，则超图$ H $被称为Berge- $ G $，使得$ e \ E（G）$中所有$ e \的subseteq f（e）$。Berge- $ G $超图族由$ \ mathcal {B}（G）$表示。在$ n $-顶点$ r $-图中没有与任何Berge- $ G $同构的亚超图的最大边数用$ ex_r（n，\ mathcal {B}（G））$表示。Gyárfás[SIAM J.Discrete Math。，33（2019），pp.383--392]显示，对于$ n \ geq 6 $，$ ex_3（n，\ mathcal {B}（K_4））= \ lfloor \ frac {n} {3} \ rfloor \ lfloor \ frac {n + 1} {3} \ rfloor \ lfloor \ frac {n + 2} {3} \ rfloor $。但是，当$ n \ ge 7 $时，我们在结果证明中发现了一个错误。Gerbner，Methuku和Palmer的最新结果[European J. Combin。，86（2020），103082]暗示对于$ n \ geq 9 $，$ ex_3（n，\ mathcal {B}（K_4））= \ lfloor \ frac {n} {3} \ rfloor \ lfloor \ frac {n + 1} {3} \ rfloor \ lfloor \ frac {n + 2} {3} \ rfloor $。在本文中，我们为证明结论的完整性，证明了其余情况$ n = 7 $和$ n = 8 $。