David Conlon (Caltech): 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 $o(n^{3/2})$ edges.- For $r \geq 3$, every $n$-vertex $r$-graph with girth greater than $5$ has $o(n^{3/2})$ edges.- Every subset of $[n]$ without a nontrivial solution to the equation $x_1 + x_2 + 2x_3 = x_4 + 3x_5$ has size $o(\sqrt{n})$.