On graphs that do not contain the cube and related problems (Q2368591)

Let \(Q\) be the edge graph of the three-dimensional cube. The Turán number of \(Q\) is the maximum number of edges in a graph on \(n\) vertices which does not contain \(Q\). \textit{P. Erdős} and \textit{M. Simonovits} [Combinat. Theory Appl. Colloquia Math. Soc. János Bolyai 4, 377--390 (1970; Zbl 0209.28001)] showed that the Turán number of \(Q\) is \(O(n^{8/5})\). The authors give an alternative simpler proof of this result. Moreover, their method gives further generalizations.