On Graphs That Do Not Contain The Cube And Related Problems

The Turan number of a graph G is the maximum number of edges in a graph on n vertices that does not contain G. For graphs G with χ(G) = r, a celebrated theorem of Erdős and Stone [2] states that the Turan number of G is ( r−2 r−1 + o (1) ) ( n 2 ) . For bipartite graphs, this result only tell us that the Turan number is o (n). The 3-dimensional cube Q is the graph with vertex set {0, 1} where two vertices are adjacent if they differ in exactly one coordinate, it can be seen that this graph is bipartite. Erdős and Simonovits [1] proved that the Turan number of Q is O ( n ) . This talk is on a paper of Pinchasi and Sharir [3] giving an alternative, simpler proof of this result. It is immediate that the following theorem is sufficient to prove the result.