Math 2.0

Combinatorics Permutations, partitions, words Theorem

Erdős–Szekeres theorem

Builds on Generalized pigeonhole principle

Statement

Let rr and ss be positive integers. Let a1,a2,…,aNa_1, a_2, \dots, a_N be a sequence of N=(r−1)(s−1)+1N = (r - 1)(s - 1) + 1 distinct real numbers. Then the sequence has an increasing subsequence of length rr or a decreasing subsequence of length ss.

Remarks

The bound is sharp: some sequence of (r−1)(s−1)(r - 1)(s - 1) distinct numbers has neither type of subsequence. For r=s=3r = s = 3: each sequence of 55 distinct numbers has a monotone subsequence of length 33. A 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 finishes the proof.

History

Paul Erdős and George Szekeres proved the theorem in 1935.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Cardinality of a finite set, definition, no evidence levelCardinality of a fini…definitionPigeonhole principle, statement, no evidence levelPigeonhole principlestatementGeneralized pigeonhole principle, statement, no evidence levelGeneralized pigeonhol…statementErdős–Szekeres theorem, statement, no evidence levelErdős–Szekeres theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cardinality of a finite set definition
  2. Pigeonhole principle statement
  3. Generalized pigeonhole principle statement
  4. Erdős–Szekeres theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Erdős–Szekeres_theorem, lead section
About this page
Tag0SNJM8HSD3
URIm2://0SNJM8HSD3
Kindstatement, theorem
WritingW1 stated
MSC05A05
StatusThis copy of the site has no build data. Run m2 status 0SNJM8HSD3.
UsesGeneralized pigeonhole principle
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