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Erdős–Szekeres theorem

A path of four positively sloped edges in a

set of 17 points. If one forms a sequence of

they-coordinates of the points, in order by

their x-coordinates, the Erdős–Szekeres

theorem ensures that there exists either a path

of this type or one the same length in w hich all

slopes are ≤ 0. How ever, if the central point is

omitted, no such path w ould exist.

From Wikipedia, the free encyclopedia

In mathematics, the Erdős–Szekeres

theorem is a finitary result that makes

precise one of the corollaries of Ramsey's

theorem. While Ramsey's theorem makes it

easy to prove that every 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 Szekeresgoes further. For

given r, s they showed that any sequence of

length at least (r − 1)(s − 1) + 1 contains 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.[1]

Contents [hide]

1 Example

2 Alternative interpretations

2.1 Geometric interpretation

2.2 Permutation pattern interpretation

3 Proofs

3.1 Pigeonhole principle

3.2 Dilworth's theorem

4 See also

5 References

6 External links

Example [edit]

For r = 3 and s = 2, the formula tells us that any permutation of three numbers has an

increasing subsequence of length three or a decreasing subsequence of length two. Among

the six permutations of the numbers 1,2,3:

1,2,3 has an increasing subsequence consisting of all three numbers

1,3,2 has a decreasing subsequence 3,2

2,1,3 has a decreasing subsequence 2,1

2,3,1 has two decreasing subsequences, 2,1 and 3,1

3,1,2 has two decreasing subsequences, 3,1 and 3,2

3,2,1 has three decreasing length-2 subsequences, 3,2, 3,1, and 2,1.

Alternative interpretations [edit]

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Geometric interpretation [edit]

One can interpret the positions of the numbers in a sequence as x-coordinates of points in

the Euclidean plane, and the numbers themselves as y-coordinates; conversely, for any point

set in the plane, the y-coordinates of the points, ordered by their x-coordinates, forms a

sequence of numbers (unless two of the points have equal x-coordinates). With this

translation between sequences and point sets, the Erdős–Szekeres theorem can be

interpreted as stating that in any set of at least rs − r − s + 2 points we can find a polygonal

path of either r − 1 positive-slope edges or s − 1 negative-slope edges. In particular

(takingr = s), in any set of at least n points we can find a polygonal path of at least ⌊√(n-1)⌋

edges with same-sign slopes. For instance, taking r = s = 5, any set of at least 17 points has

a four-edge path in which all slopes have the same sign.

An example of rs − r − s + 1 points without such a path, showing that this bound is tight, can

be formed by applying a small rotation to an (r − 1) by (s − 1) grid.

Permutation pattern interpretation [edit]

The Erdős–Szekeres theorem may also be interpreted in the language of permutation

patterns as stating that every permutation of length at least rs + 1 must contain either the

pattern 1, 2, 3, ..., r + 1 or the pattern s + 1, s, ..., 2, 1.

Proofs [edit]

The Erdős–Szekeres theorem can be proved in several different ways; Steele (1995) surveys

six different proofs of the Erdős–Szekeres theorem, including the following two.[2] Other

proofs surveyed by Steele include the original proof by Erdős and Szekeres as well as those

of Blackwell (1971),[3] Hammersley (1972),[4] and Lovász (1979).[5]

Pigeonhole principle [edit]

Given a sequence of length (r − 1)(s − 1) + 1, label each number ni in the sequence with the

pair (ai,b i), where ai is the length of the longest monotonically increasing subsequence

ending with ni and b i is the length of the longest monotonically decreasing subsequence

ending with ni. Each two numbers in the sequence are labeled with a different pair:

if i < j andni < nj then ai < aj, and on the other hand if ni > nj then b i < b j. But there are only

(r − 1)(s − 1) possible labels in which ai is at most r − 1 and b i is at most s − 1, so by

thepigeonhole principle there must exist a value of i for which ai or b i is outside this range.

If ai is out of range then ni is part of an increasing sequence of length at least r, and if b i is

out of range then ni is part of a decreasing sequence of length at least s.

Steele (1995) credits this proof to the one-page paper of Seidenberg (1959) and calls it "the

slickest and most systematic" of the proofs he surveys.[2][6]

Dilworth's theorem [edit]

Another of the proofs uses Dilworth's theorem on chain decompositions in partial orders, or

its simpler dual (Mirsky's theorem).

To prove the theorem, define a partial ordering on the members of the sequence, in which x is

less than or equal to y in the partial order if x ≤ y as numbers and x is not later than y in the

sequence. A chain in this partial order is a monotonically increasing subsequence, and

anantichain is a monotonically decreasing subsequence. By Mirsky's theorem, either there is

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a chain of length r, or the sequence can be partitioned into at most r − 1 antichains; but in

that case the largest of the antichains must form a decreasing subsequence with length at

least

Alternatively, by Dilworth's theorem itself, either there is an antichain of length s, or the

sequence can be partitioned into at most s − 1 chains, the longest of which must have length

at least r.