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  "body": "\n## Statement\n\nIn arithmetic combinatorics, the **corners theorem** states that for every $\\varepsilon\u003e0$, for large enough $N$, any set of at least $\\varepsilon N^2$ points in the $N\\times N$ grid $\\{1,\\ldots,N\\}^2$ contains a corner, i.e., a triple of points of the form $\\{(x,y), (x+h,y), (x,y+h)\\}$ with $h\\ne 0$. It was first proved by Miklós Ajtai and Endre Szemerédi in 1974 using Szemerédi's theorem. In 2003, József Solymosi gave a short proof using the triangle removal lemma.\n",
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