The Induced Removal Lemma in Sparse Graphs

The induced removal lemma of Alon, Fischer, Krivelevich and Szegedy states that if an n-vertex graph G is ε-far from being induced H-free then G contains δ_H(ε) · n^h induced copies of H. Improving upon the original proof, Conlon and Fox proved that 1/δ_H(ε) is at most a tower of height poly(1/ε), and asked if this bound can be further improved to a tower of height log (1/ε). In this paper we obtain such a bound for graphs G of density O(ε). 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.