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.

In this note, we give a counter example to show that the proof of a main result obtained by Haynes and Slater (Networks 32 (1998), 199–206, Theorem 12) is inaccurate. Here, we give a complete proof of the result. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 115–116 2011