48 research outputs found
A combinatorial proof of the Removal Lemma for Groups
Green [Geometric and Functional Analysis 15 (2005), 340--376] established a
version of the Szemer\'edi Regularity Lemma for abelian groups and derived the
Removal Lemma for abelian groups as its corollary. We provide another proof of
his Removal Lemma that allows us to extend its statement to all finite groups.
We also discuss possible extensions of the Removal Lemma to systems of
equations
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