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  "body": "\n## Statement\n\nFor every real number $\\varepsilon \u003e 0$ there is a real number $\\delta \u003e 0$ with the following property.\nLet $G$ be a finite simple [[graph|00C0BR7D7H]] with $n$ vertices that contains at most $\\delta n^3$ triangles.\nThen one can delete at most $\\varepsilon n^2$ edges of $G$ so that the resulting graph has no triangle.\n\n## Remarks\n\nThe standard proof applies the [[Szemerédi regularity lemma|ETMAPNQY7T]] with a small accuracy.\nDelete the edges inside parts, between irregular pairs, and between pairs of low density; these are at most $\\varepsilon n^2$ edges.\nIf a triangle survives, its three parts are pairwise regular and dense, and the counting lemma then gives more than $\\delta n^3$ triangles.\nThe resulting $\\delta$ is very small: the proof gives $1/\\delta$ of tower type in $1/\\varepsilon$.\n\nA consequence: if every edge of a graph on $n$ vertices lies in exactly one triangle, then the graph has $o(n^2)$ edges.\nIndeed, if it has $e$ edges, it has $e/3 \\le n^2/6$ triangles, which is less than $\\delta n^3$ for large $n$.\nThe triangles are edge-disjoint, so destroying them needs $e/3$ deletions, and the lemma gives $e/3 \\le \\varepsilon n^2$.\nApplied to a graph built from a subset of $\\Z/N\\Z$, this gives [[Roth's theorem|E7H8NWXESF]] on three-term arithmetic progressions.\n\n## Explanations\n\n### L1 human\n\nA graph with few triangles, compared with the $n^3$ triangles that are possible, is close to a graph with no triangles at all.\n\"Close\" means that a small fraction of the $n^2$ edges is enough to destroy every triangle.\nThis sounds obvious but is not: the few triangles could share edges in a clever way, and the lemma says they cannot.\n\n### L2 human\n\nThe lemma is the simplest case of the graph removal lemma.\nThere, any fixed graph $H$ takes the place of the triangle, and $\\delta n^{|V(H)|}$ copies take the place of $\\delta n^3$ triangles.\nIt is a central tool linking [[extremal graph theory|SC5QB5F4TE]] and [[additive combinatorics|3NM362A2JG]].\nIts use for Roth's theorem shows how a statement about graphs can prove a statement about integers.\nBetter bounds for $\\delta$ than tower type in this generality are a well-known open question.\n",
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