For a positive integer t, let F_t denote the graph of the t by t grid. Motivated by a 50-year-old conjecture of Erdos about Turan numbers of r-degenerate graphs, we prove that there exists a  constant C=C(t) such that ex(n,F_t) < Cn^{3/2}. This bound is tight up to the value of C. Our original proof of this result relied on an intricate argument using the tensor power trick. In this talk, I will present a simplified version of the proof.

Based on joint work with Oliver Janzer, Benny Sudakov and Istvan Tomon.