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  "body": "\n## Problem\n\nA finite set of points in the plane is in general position if no three of its points lie on a line.\nFor an integer $n \\ge 3$, let $N(n)$ be the least integer $N$ with the following property.\nEvery set of $N$ points in the plane in general position contains $n$ points that are the vertices of a [[convex|02AS45T3H9]] polygon.\nThe problem asks for the value of $N(n)$.\nErdős and Szekeres conjectured that $N(n) = 2^{n-2} + 1$ for every $n \\ge 3$.\n\n## Remarks\n\nErdős and Szekeres proved that $N(n)$ exists, using [[Ramsey's theorem|00TQV3F5S4]] in one of their proofs.\nThey also proved the lower bound $N(n) \\ge 2^{n-2} + 1$ by a construction.\nThe known values are $N(3) = 3$, $N(4) = 5$, $N(5) = 9$ and $N(6) = 17$, all equal to $2^{n-2} + 1$.\nThe value $N(6) = 17$ was established with a computer search by Szekeres and Peters.\nSuk proved in 2017 that $N(n) \\le 2^{n + o(n)}$.\nThe conjecture $N(n) = 2^{n-2} + 1$ is open for $n \\ge 7$.\n\n## History\n\nThe case $n = 4$ was posed by Esther Klein.\nThe name comes from the marriage of Klein and George Szekeres, who met through this problem.\n\n## Explanations\n\n### L1 human\n\nPut five points on a sheet of paper so that no three are on one straight line.\nThen four of them always form a convex quadrilateral, a shape with no dent.\nThe problem asks how many points you need to be sure of finding a convex shape with $n$ corners.\nThe guess is $2^{n-2} + 1$ points, and it is proved only for small $n$.\n",
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