# Ramsey Theory, integer partitions and a new proof of the Erdős–Szekeres Theorem

@article{Moshkovitz2012RamseyTI, title={Ramsey Theory, integer partitions and a new proof of the Erdős–Szekeres Theorem}, author={Guy Moshkovitz and Asaf Shapira}, journal={Advances in Mathematics}, year={2012}, volume={262}, pages={1107-1129} }

Abstract Let H be a k -uniform hypergraph whose vertices are the integers 1 , … , N . We say that H contains a monotone path of length n if there are x 1 x 2 ⋯ x n + k − 1 so that H contains all n edges of the form { x i , x i + 1 , … , x i + k − 1 } . Let N k ( q , n ) be the smallest integer N so that every q -coloring of the edges of the complete k -uniform hypergraph on N vertices contains a monochromatic monotone path of length n . While the study of N k ( q , n ) for specific values of k… Expand

#### 41 Citations

Variants of the Erdős-Szekeres and Erdős-Hajnal Ramsey problems

- Mathematics, Computer Science
- Eur. J. Comb.
- 2017

An ordered version of the classical Erdős–Hajnal hypergraph Ramsey problem is considered, the tower height given by the trivial upper bound is improved, and it is conjecture that this tower height is optimal. Expand

The Erdős–Hajnal hypergraph Ramsey problem

- Mathematics
- 2016

Given integers 2 ≤ t ≤ k+1 ≤ n, let gk(t, n) be the minimum N such that every red/blue coloring of the k-subsets of {1, . . . , N} yields either a (k + 1)-set containing t red k-subsets, or an n-set… Expand

Higher-order Erdős–Szekeres theorems

- Mathematics
- 2013

Abstract Let P = ( p 1 , p 2 , … , p N ) be a sequence of points in the plane, where p i = ( x i , y i ) and x 1 x 2 ⋯ x N . A famous 1935 Erdős–Szekeres theorem asserts that every such P contains a… Expand

Ordered Ramsey numbers of loose paths and matchings

- Computer Science, Mathematics
- Discret. Math.
- 2016

It is determined that the ordered Ramsey numbers of loose paths under a monotone order grows as a tower of height two less than the maximum degree in terms of the number of edges. Expand

Ordered Ramsey numbers

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 2017

It is proved that even for matchings there are labelings where the ordered Ramsey number is superpolynomial in the number of vertices and a general upper bound on ordered Ramsey numbers is proved. Expand

Two extensions of the Erdős–Szekeres problem

- Mathematics
- 2017

According to Suk’s breakthrough result on the Erdős–Szekeres problem, any point set in general position in the plane, which has no n elements that form the vertex set of a convex n-gon, has at most… Expand

A note on order-type homogeneous point sets

- Mathematics, Computer Science
- ArXiv
- 2013

It is shown that OT_3(n) = 2^(2^(Theta(n)), answering a question of Eli\'a and Matou\v{s}ek, and new bounds are given for OT_d(n), which is bounded above by an exponential tower of height d with O(n). Expand

Increasing Hamiltonian paths in random edge orderings

- Computer Science, Mathematics
- Random Struct. Algorithms
- 2016

The surprising result that in the random setting, S(f) often takes its maximum possible value of n – 1 (visiting all of the vertices with an increasing Hamiltonian path) is discovered, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely. Expand

Hamiltonian increasing paths in random edge orderings

- Mathematics
- 2014

If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained… Expand

Two extensions of the Erd\H{o}s-Szekeres problem

- Mathematics
- 2017

According to Suk's breakthrough result on the Erdos-Szekeres problem, any point set in general position in the plane, which has no $n$ elements that form the vertex set of a convex $n$-gon, has at… Expand

#### References

SHOWING 1-10 OF 31 REFERENCES

Erdos-Szekeres-type theorems for monotone paths and convex bodies

- Mathematics
- 2012

For any sequence of positive integers j(1) = 2 and q >= 2, what is the smallest integer N with the property that no matter how we color all k-element subsets of [N]={1, 2, ..., N} with q colors, we… Expand

Higher-order Erdös: Szekeres theorems

- Computer Science, Mathematics
- SoCG '12
- 2012

An Ω(log(k-1)N) lower bound is obtained based on a quantitative Ramsey-type theorem for what the authors call transitive colorings of the complete (k+1)-uniform hypergraph; it provides a unified view of the two classical Erdos-Szekeres results. Expand

Weyl Groups, the Hard Lefschetz Theorem, and the Sperner Property

- Mathematics, Computer Science
- SIAM J. Algebraic Discret. Methods
- 1980

Techniques from algebraic geometry are used to show that certain finite partially ordered sets Q^X derived from a class of algebraic varieties X have the k-Sperner property for all k, which means that there is a simple description of the cardinality of the largest subset of $Q^X $ containing no $( k + 1 )$-element chain. Expand

The Erdos-Szekeres problem on points in convex position – a survey

- Mathematics
- 2000

In 1935 Erdős and Szekeres proved that for any integer n ≥ 3 there exists a smallest positive integer N(n) such that any set of at least N(n) points in general position in the plane contains n points… Expand

On Dedekind’s problem: the number of isotone Boolean functions. II

- Mathematics
- 1975

It is shown that 0(n), the size of the free distributive lattice on n generators (which is the number of isotone Boolean functions on subsets of an n element set), satisfies [n1 i (n) < 2(1 +0(1og… Expand

Entropy, independent sets and antichains: A new approach to Dedekind's problem

- Mathematics
- 2001

For n-regular, N-vertex bipartite graphs with bipartition A U B, a precise bound is given for the sum over independent sets I of the quantity μ |I ∩ A| λ |I ∩ B| , (In other language, this is… Expand

Some asymptotic formulas for lattice paths

- Mathematics
- 1986

Abstract Denote by ϱ( n,j ), ϱ( a,b,j ) and w ( n,j ) the numbers of two-dimensional lattice paths satisfying respectively the following three conditions: (i) The paths have length n and the area… Expand

On the set of divisors of a number

- Mathematics
- 1951

If z is a natural number and if z=pipfy —Pj is its factorization into primes, then the sum X/ + \2 + '" + \ " will be called the degree of z. Let m be a squarefree natural number of degree /?, i.e.,… Expand

Memoir on the Theory of the Partition of Numbers. Part I

- Mathematics
- 1896

Art. 1. I have under consideration multipartite numbers as defined in a former paper. I recall that the multipartite number ᾱβγ . . . , may be regarded as specifying α + β + γ . . . things, α of one… Expand

On Dedekind’s problem: The number of monotone Boolean functions

- Mathematics
- 1969

with an = ce~nli, /3„ = e'(Iog w)/«1'2.The number \p(n) is equal to the number of ideals, or of antichains,or of monotone increasing functions into 0 and 1 definable on thelattice of subsets of an… Expand