Induced Graph Removal Lemma

Statement

A natural generalization of the graph removal lemma is to consider induced subgraphs. In property testing, it is often useful to consider how far a graph is from being induced $H$-free. A graph $G$ is considered to contain an induced subgraph $H$ if there is an injective map $f: V(H) \rightarrow V(G)$ such that $(f(u),f(v))$ is an edge of $G$ if and only if $(u,v)$ is an edge of $H$. Notice that non-edges are considered as well. $G$ is induced $H$-free if there is no induced subgraph $G$. We define $G$ as $\epsilon$-far from being induced $H$-free if we cannot add or delete $\epsilon n^2$ edges to make $G$ induced $H$-free.