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  "body": "\n## Statement\n\nLet $r$ and $s$ be positive integers.\nLet $a_1, a_2, \\dots, a_N$ be a sequence of $N = (r - 1)(s - 1) + 1$ distinct real numbers.\nThen the sequence has an increasing subsequence of length $r$ or a decreasing subsequence of length $s$.\n\n## Remarks\n\nThe bound is sharp: some sequence of $(r - 1)(s - 1)$ distinct numbers has neither type of subsequence.\nFor $r = s = 3$: each sequence of $5$ distinct numbers has a monotone subsequence of length $3$.\nA short proof labels each term with the lengths of the longest increasing and decreasing subsequences that end at it. The labels are distinct, and the [[pigeonhole principle|0HGVCWQGY3]] finishes the proof.\n\n## History\n\nPaul Erdős and George Szekeres proved the theorem in 1935.\n\n## Explanations\n\n### L1 human\n\nA long enough list of different numbers always contains a long subsequence that goes up or one that goes down. The terms need not be adjacent.\nFor example, $3, 1, 4, 5, 2$ has the increasing subsequence $3, 4, 5$.\n",
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