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 Szekeres goes 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.
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.
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. 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.
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.
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. Other proofs surveyed by Steele include the original proof by Erdős and Szekeres as well as those of Blackwell (1971), Hammersley (1972), and Lovász (1979).
Given a sequence of length (r − 1)(s − 1) + 1, label each number ni in the sequence with the pair (ai,bi), where ai is the length of the longest monotonically increasing subsequence ending with ni and bi 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 and ni < nj then ai < aj, and on the other hand if ni > nj then bi < bj. But there are only (r − 1)(s − 1) possible labels in which ai is at most r − 1 and bi is at most s − 1, so by the pigeonhole principle there must exist a value of i for which ai or bi 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 bi is out of range then ni is part of a decreasing sequence of length at least s.
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 an antichain is a monotonically decreasing subsequence. By Mirsky's theorem, either there is 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.
- Erdős, Paul; Szekeres, George (1935), "A combinatorial problem in geometry", Compositio Mathematica 2: 463–470.
- Steele, J. Michael (1995), "Variations on the monotone subsequence theme of Erdős and Szekeres", in Aldous, David; Diaconis, Persi; Spencer, Joel et al., Discrete Probability and Algorithms, IMA Volumes in Mathematics and its Applications 72, Springer-Verlag, pp. 111–131 .
- Blackwell, Paul (1971), "An alternative proof of a theorem of Erdős and Szekeres", American Mathematical Monthly 78 (3): 273–273, doi:10.2307/2317525, JSTOR 2317525.
- Hammersley, J. M. (1972), "A few seedlings of research", Proc. 6th Berkeley Symp. Math. Stat. Prob., University of California Press, pp. 345–394. As cited by Steele (1995).
- Lovász, László (1979), "Solution to Exercise 14.25", Combinatorial Problems and Exercises, North-Holland. As cited by Steele (1995).
- Seidenberg, A. (1959), "A simple proof of a theorem of Erdős and Szekeres", Journal of the London Mathematical Society 34: 352.