844 research outputs found
Graph removal lemmas
The graph removal lemma states that any graph on n vertices with o(n^{v(H)})
copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite
its innocent appearance, this lemma and its extensions have several important
consequences in number theory, discrete geometry, graph theory and computer
science. In this survey we discuss these lemmas, focusing in particular on
recent improvements to their quantitative aspects.Comment: 35 page
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 Szemer\'edi'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.Comment: 17 page
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