A new proof of the graph removal lemma

Let H be a fixed graph with h vertices. The graph removal lemma states that every graph on n vertices with o(n h ) copies of H can be made H-free by removing o(n 2 ) edges. We give a new proof which avoids Szemerdi's regularity lemma and gives a better bound. This approach also works to give improved bounds for the directed and multicolored analogues of the graph removal lemma. This answers questions of Alon and Gowers.