Complete bipartite Turan numbers

Let H be a graph. The function ex(n,H) is the maximum number of edges that a graph with n vertices can have, which contains no subgraph isomorphic to H.

If H is not bipartite then the asymptotic behaviour of ex(n,H) is known, but if H is bipartite then in general this is not the case. This talk will focus on the case that H is a complete bipartite graph. I will review the previous constructions from a geometrical point of view and explain how this enables us to improve the lower bound of ex(n,K_5,5).