Related papers: A positive fraction Erdos-Szekeres theorem and its…

The classical Erd\H{o}s-Szekeres theorem dating back almost a hundred years states that any sequence of $(n-1)^2+1$ distinct real numbers contains a monotone subsequence of length $n$. This theorem has been generalised to higher dimensions…

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 Erdos--Szekeres theorem asserts that every such P contains a monotone subsequence S of $\sqrt N$ points. Another, equally…

One of the most classical results in Ramsey theory is the theorem of Erd\H{o}s and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following…

This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erd\H{o}s-Szekeres theorem: For every $k\ge1$, every order-$n$ $k$-dimensional permutation contains a monotone subsequence of…

A $k$-modal sequence is a sequence of real numbers that can be partitioned into $k+1$ (possibly empty) monotone sections such that adjacent sections have opposite monotonicities. For every positive integer $k$, we prove that any sequence of…

For any sequence of positive integers j_1 < j_2 < ... < j_n, the k-tuples (j_i,j_{i + 1},...,j_{i + k-1}), i=1, 2,..., n - k+1, are said to form a monotone path of length n. Given any integers n\ge k\ge 2 and q\ge 2, what is the smallest…

For any integer $n \geq 2$, let $(m_{1},\ldots,m_{n})$ be a strictly increasing $n$-tuple of positive integers. We show that any subset $A\subset [N]^n$ of density at least $(\log N)^{-c}$ contains a nontrivial configuration of the form…

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…

A classical and widely used lemma of Erdos and Szekeres asserts that for every n there exists N such that every N-term sequence a of real numbers contains an n-term increasing subsequence or an n-term nondecreasing subsequence;…

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general…

Erd\H{o}s and Szekeres showed that given a permutation $p$ of $[n]$, and the sequence defined by \newline $(p(1), p(2), \ldots, p(n))$, there exists either a decreasing or increasing subsequence, not necessarily contiguous, of length at…

Szemer\'edi's Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain…

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of…

According to the Erd\H{o}s-Szekeres theorem, for every $n$, a sufficiently large set of points in general position in the plane contains $n$ in convex position. In this note we investigate the line version of this result, that is, we want…

The investigation of primes in certain arithmetic sequences is one of the fundamental problems in number theory and especially, finding blocks of distinct primes has gained a lot of attention in recent years. In this context, we prove the…

The Erd\H{o}s-Szekeres Theorem stated in terms of graphs says that any red-blue coloring of the edges of the ordered complete graph $K_{rs+1}$ contains a red copy of the monotone increasing path with $r$ edges or a blue copy of the monotone…

A famous theorem of Szemer\'edi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the…

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a…

A strictly increasing sequence of positive integers is called a slightly curved sequence with small error if the sequence can be well-approximated by a function whose second derivative goes to zero faster than or equal to $1/x^\alpha$ for…

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemer\'edi theorem which says that any subset of a pseudorandom set of…