Any sequence of more than (r minus 1) times (s minus 1) distinct real numbers contains either an increasing subsequence of length r or a decreasing subsequence of length s. Proved by Paul Erdos and George Szekeres, it is a classical result connecting Ramsey-type combinatorics to the structure of sequences, originally developed in their work on the 'happy ending problem' for points in general position.
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StatementAny sequence of distinct real numbers with length at least (r-1)(s-1)+1 contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s. 1 Classification
Statement Form Erdos-Szekeres Theorem
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Source Erdos-Szekeres Theorem (Wikipedia)
Source Erdos-Szekeres Theorem (Wikipedia)
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1. Erdos-Szekeres Theorem (Wikipedia)
Wikimedia FoundationLead section
contains a monotonically increasing subsequence of length r or a monotonically decreasing subsequence of length s.
Lead section, proof year
The proof appeared in the same 1935 paper that mentions the Happy Ending problem.
View the Source Erdos-Szekeres Theorem (Wikipedia)
Principal Result Of: Paul Erdos, Lead sentence of the article on the result
While Ramsey's theorem makes it easy to prove that every infinite sequence of distinct real numbers contains a monotonically increasing infinite subsequence or a monotonically decreasing infinite subsequence, the result proved by Paul Erdős and George Szekeres goes further.
Principal Result Of: George Szekeres, Lead sentence of the article on the result
While Ramsey's theorem makes it easy to prove that every infinite sequence of distinct real numbers contains a monotonically increasing infinite subsequence or a monotonically decreasing infinite subsequence, the result proved by Paul Erdős and George Szekeres goes further.
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