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  "body": "\n## Proof\n\nWith these results, we are able to prove the induced graph removal lemma. Take any graph $G$ with $n$ vertices that has less than $\\delta n^{v(H)}$ copies of $H$. The idea is to start with a collection of vertex sets $W_i$ which satisfy the conditions of the **Corollary of the Strong Regularity Lemma**. We then can perform a \"cleaning\" process where we remove all edges between pairs of parts $(W_i,W_j)$ with low density, and we can add all edges between pairs of parts $(W_i,W_j)$ with high density. We choose the density requirements such that we added/deleted at most $\\epsilon n^2$ edges.\n\nIf the new graph has no copies of $H$, then we are done. Suppose the new graph has a copy of $H$. Suppose the vertex $v_i \\in v(H)$ is embedded in $W_{f(i)}$. Then if there is an edge connecting $v_i,v_j$ in $H$, then $W_i,W_j$ does not have low density. (Edges between $W_i,W_j$ were not removed in the cleaning process.) Similarly, if there is not an edge connecting $v_i,v_j$ in $H$, then $W_i,W_j$ does not have high density. (Edges between $W_i,W_j$ were not added in the cleaning process.)\n\nThus, by a similar counting argument to the proof of the triangle counting lemma (that is, the **graph counting lemma**), we can show that $G$ has more than $\\delta n^{v(H)}$ copies of $H$.\n",
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