The regularity method for graphs with few 4-cycles

We develop a sparse graph regularity method that applies to graphs with few 4-cycles, including new counting and removal lemmas for 5-cycles in such graphs. Some applications include: Every n-vertex graph with no 5-cycle can be made triangle-free by deleting o(n3/2) edges. For r > 3, every n-vertex r-graph with girth greater than 5 has o(n3/2) edges. Every subset of [n] without a nontrivial solution to the equation x1+x2+2x3=x4+3x5 has size o(n).