An Arithmetic Analogue of Fox's Triangle Removal Argument

We give an arithmetic version of the recent proof of the triangle removal lemma by Fox [Fox11], for the group F n . A triangle in F n is a tuple (x,y,z) such that x + y + z = 0. The triangle removal lemma for F n states that for every " > 0 there is a � > 0, such that if a subset A of F n requires the removal of at least " · 2 n elements to make it triangle-free, then it must contain at least � · 2 2n triangles. We give a direct proof which gives an improved lower bound for � (as a function of "), analogous to the one obtained by Fox for triangle removal in graphs. The improved lower bound was already known to follow (for triangle-removal in all groups), ′ |

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