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.

Summary In recent years, with the prevalence of smartphones, the number of Android malware shows explosive growth. As malicious apps may steal users' sensitive data and even money from mobile and bank accounts, it is important to detect potential malicious behaviors so as to block them. To achieve this goal, we propose a dynamic behavior inspection and analysis framework for malicious behavior detection. A customized Android system is built to record apps' API calls, permission uses, and some other runtime features. We also develop an automated app behavior inspection platform to install and inspect massive samples so as to collect apps' dynamic behavior records. Then these records are exploited to train a string subsequence kernel–based Support Vector Machine (SVM) model, which can be used to classify benign and malicious behaviors offline. To realize online detection, we further extract apps' runtime features including sensitive permission combination uses, sensitive behavior sequences, and user interactions for behavior classification. The classification results can reach an accuracy of 84.9% in offline phase and 99.0% in online phase. Besides, we verify our scheme for identifying malicious apps, and the results show that 71.8% instances of malware samples are identified by running each app for only 18 minutes.