The Induced Removal Lemma in Sparse Graphs

The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if ann-vertex graphGis epsilon-far from being inducedH-free thenGcontains delta(H)(epsilon) center dot n(h)induced copies ofH. Improving upon the original proof, Conlon and Fox proved that 1/delta(H)(epsilon)is at most a tower of height poly(1/epsilon), and asked if this bound can be further improved to a tower of height log(1/epsilon). In this paper we obtain such a bound for graphsGof densityO(epsilon). We actually prove a more general result, which, as a special case, also gives a new proof of Fox's bound for the (non-induced) removal lemma.