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

TL;DR

This paper provides a new proof of a theorem by Erdős and Simonovits on the maximum number of edges in graphs that do not contain a cube as a subgraph, including bipartite variants.

Contribution

It offers a self-contained proof of the Turán number for the cube and discusses related bipartite cases, improving understanding of cube-free graphs.

Findings

e(G) < n^{8/5} + (2n)^{3/2} for cube-free graphs

Provides a self-contained proof of Erdős-Simonovits theorem

Discusses bipartite versions of the problem

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.