The removal lemma is a cornerstone of extremal graph theory. It has many generalizations and extensions to other combinatorial structures, such as induced subgraphs, hypergraphs, digraphs and matrices. A general open question is to characterize the cases in which the removal lemma has polynomial bounds. I will survey the state of knowledge on this general problem and present some new results.
Based on joint works with A. Shapira and I. Tomon.