On a theorem of Erd\H{o}s and Simonovits on graphs not containing the cube

Abstract

The cube Q is the usual 8-vertex graph with 12 edges. Here we give a new proof for a theorem of Erd\H{o}s and Simonovits concerning the Tur\'an number of the cube. Namely, it is shown that e(G) < n^{8/5}+(2n)^{3/2} holds for any n-vertex cube-free graph G. Our aim is to give a self-contained exposition. We also point out the best known results and supply bipartite versions.