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  "body": "\n## Problem\n\nLet $n \\ge 3$ and let $P$ be a set of $n$ points in the unit square $[0, 1]^2$.\nLet $\\delta(P)$ be the smallest area of a [[triangle|0CTNFGYMM2]] whose vertices are three points of $P$.\nThree collinear points count as a triangle of area $0$.\nLet $\\Delta(n)$ be the supremum of $\\delta(P)$ over all such $P$.\nDetermine the order of growth of $\\Delta(n)$ as $n \\to \\infty$.\n\n## Remarks\n\nHans Heilbronn conjectured that $\\Delta(n) = O(1/n^2)$, with $O$ the [[big O notation|0RYTGBJAGN]]; the problem first appeared in print in a 1951 paper of Klaus Roth.\nErdős showed $\\Delta(n) \\ge c/n^2$ for a constant $c \u003e 0$, so $1/n^2$ would be the right order.\nKomlós, Pintz and Szemerédi disproved the conjecture in 1982 by showing $\\Delta(n) \\ge c \\log n / n^2$.\nOn the other side, Roth proved $\\Delta(n) = o(1/n)$, and Komlós, Pintz and Szemerédi proved $\\Delta(n) \\le n^{-8/7 + o(1)}$ in 1981.\nIn 2023 Cohen, Pohoata and Zakharov improved the exponent $-8/7$ for the first time.\nThe exact order of growth remains open.\n\n## Explanations\n\n### L1 human\n\nScatter $n$ points in a square so that every triangle formed by three of them has a large area.\nSince there are many points, some triangle must be small; the question is how small.\nThe answer lies somewhere between about $(\\log n)/n^2$ and a little below $n^{-8/7}$, and nobody knows exactly where.\n",
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      "first_stated": "1951",
      "posed_by": "Hans Heilbronn"
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      "52C10",
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    "name": "Heilbronn triangle problem",
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      "wikipedia": "Heilbronn triangle problem"
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        "url": "https://en.wikipedia.org/wiki/Heilbronn_triangle_problem"
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  "msc": [
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  "name": "Heilbronn triangle problem",
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