High Quality Content by WIKIPEDIA articles! In mathematics, the Erd?s–Szekeres theorem is a finitary result, which makes precise one of the corollaries of Ramsey's theorem. While Ramsey's theorem makes it easy to prove that any sequence of distinct real numbers contains either a monotonically increasing infinite subsequence, or a monotonically decreasing infinite subsequence, the result proved by Paul Erd?s and George Szekeres goes further. For given r, s they showed that any sequence of length at least (r ? 1)(s ? 1) + 1 contains either a monotonically increasing subsequence of length r, or a monotonically decreasing subsequence of length s. The proof appeared in the same 1935 paper that mentions the Happy Ending problem. Steele (1995) contains "six or more" proofs of the theorem.

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