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  "body": "\n## Statement\n\nIf we have enough information about the regularity of a graph, we can count the number of copies of a specific subgraph within the graph up to small error.\n\n\u003e **Graph Counting Lemma.** Let $H$ be a graph with $V(H)=[k]$, and let $\\varepsilon\u003e0$. Let $G$ be an $n$-vertex graph with vertex sets $V_1,\\dots,V_k\\subseteq V(G)$ such that $(V_i,V_j)$ is $\\varepsilon$-regular whenever $\\{i,j\\}\\in E(H)$. Then, the number of labeled copies of $H$ in $G$ is within $e(H)\\varepsilon|V_1|\\cdots|V_k|$ of\n\u003e\n\u003e $$\\left(\\prod_{\\{i,j\\}\\in E(H)}d(V_i,V_j)\\right)\\left(\\prod_{i=1}^k|V_i|\\right).$$\n\nThis can be combined with Szemerédi's regularity lemma to prove the Graph removal lemma. The graph removal lemma can be used to prove Roth's Theorem on Arithmetic Progressions, and a generalization of it, the hypergraph removal lemma, can be used to prove Szemerédi's theorem.\n\nThe graph removal lemma generalizes to [[induced subgraphs|0EREPBVG2G]], by considering edge edits instead of only edge deletions. This was proved by Alon, Fischer, Krivelevich, and Szegedy in 2000. However, this required a stronger variation of the regularity lemma.\n\nSzemerédi's regularity lemma does not provide meaningful results in sparse graphs. Since sparse graphs have subconstant edge densities, $\\varepsilon$-regularity is trivially satisfied. Even though the result seems purely theoretical, some attempts  have been made to use the regularity method as compression technique for large graphs.\n",
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