Regularity methods for sparse graphs and its applications

The extension of Szemerédi's regularity method to sparse settings has led to important applications, with the Green-Tao theorem being a notable example.

We develop new counting and removal lemmas for graphs with few 4-cycles. I will present a number of applications to extremal problems on graphs, hypergraphs, and sets avoiding solutions to certain equations.

Joint work with David Conlon, Jacob Fox, and Benny Sudakov.