Abstract
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 Szemerédi’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.

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ISSN = {00973165},
CODEN = {JCBTA7},
MRCLASS = {05C35 (05C65 05C75 11B75 37A45)},
MRNUMBER = {2259060},
MRREVIEWER = {Jozef Skokan},
DOI = {10.1016/j.jcta.2005.11.006},
ZBLNUMBER = {1105.05052},
} 
[Ta1] T. Tao, Structure and Randomness: Pages from Year One of a Mathematical Blog, Providence, RI: Amer. Math. Soc., 2008.
@book {Ta1, MRKEY = {2459552},
AUTHOR = {Tao, Terence},
TITLE = {Structure and Randomness: Pages from Year One of a Mathematical Blog},
PUBLISHER = {Amer. Math. Soc.},
ADDRESS = {Providence, RI},
YEAR = {2008},
PAGES = {xii+298},
ISBN = {9780821846957},
MRCLASS = {0002},
MRNUMBER = {2459552},
MRREVIEWER = {W. T. Gowers},
ZBLNUMBER = {05380664},
}