Related papers: Higher-order Erdos--Szekeres theorems

Let $S \subset \mathbb{R}^n$ have size $|S| > \ell^{2^n-1}$. We show that there are distinct points $\{x^1,..., x^{\ell+1}\} \subset S$ such that for each $i \in [n]$, the coordinate sequence $(x^j_i)_{j=1}^{\ell+1}$ is strictly increasing,…

We show that Ramsey theory, a domain presently conceived to guarantee the existence of large homogeneous sets for partitions on k-tuples of words (for every natural number k) over a finite alphabet, can be extended to one for partitions on…

Given $s \ge k\ge 3$, let $h^{(k)}(s)$ be the minimum $t$ such that there exist arbitrarily large $k$-uniform hypergraphs $H$ whose independence number is at most polylogarithmic in the number of vertices and in which every $s$ vertices…

The determination of $ES(7)$ is the first open case of the planar Erd\H{o}s--Szekeres problem, where the general conjecture predicts $ES(7)=33$. We present a SAT encoding for the 33-point case based on triple-orientation variables and a…

Given two vertex-ordered graphs $G$ and $H$, the ordered Ramsey number $R_<(G,H)$ is the smallest $N$ such that whenever the edges of a vertex-ordered complete graph $K_N$ are red/blue-coloured, then there is a red (ordered) copy of $G$ or…

We develop a new technique that allows us to show in a unified way that many well-known combinatorial theorems, including Tur\'an's theorem, Szemer\'edi's theorem and Ramsey's theorem, hold almost surely inside sparse random sets. For…

For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$…

A $k$-ordering of a graph $G$ assigns distinct order-labels from the set $\{1,\ldots,|G|\}$ to $k$ vertices in $G$. Given a $k$-ordering $H$, the ordered Ramsey number $R_<(H)$ is the minimum $n$ such that every edge-2-coloring of the…

According to a classical result of Szemer\'{e}di, every dense subset of $1,2,...,N$ contains an arbitrary long arithmetic progression, if $N$ is large enough. Its analogue in higher dimensions due to F\"urstenberg and Katznelson says that…

The induced Ramsey number $R_{\mathrm{ind}}(H; r)$ of a graph $H$ is the minimum number $N$ such that there exists a graph with $N$ vertices for which all $r$-colourings of its edges contain a monochromatic induced copy of $H$. Our main…

A path $v_1,v_2,\ldots,v_m$ in a graph $G$ is $degree$-$monotone$ if $deg(v_1) \leq deg(v_2) \leq \cdots \leq deg(v_m)$ where $deg(v_i)$ is the degree of $v_i$ in $G$. Longest degree-monotone paths have been studied in several recent…

We establish the following two main results on order types of points in general position in the plane (realizable simple planar order types, realizable uniform acyclic oriented matroids of rank $3$): (a) The number of extreme points in an…

This paper investigates several classical and novel variations of the Erd\H{o}s--Szekeres problem, including multicolored point sets, convex hexagons with a given number of interior points, and polygons with constraints on edge colors. We…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

For a finite point set $E\subset \mathbb{R}^d$ and a connected graph $G$ on $k+1$ vertices, we define a $G$-framework to be a collection of $k + 1$ points in E such that the distance between a pair of points is specified if the…

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy…

Let $P$ be a set of $n$ points in the plane. We consider a variation of the classical Erd\H{o}s-Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset $S$ of $P$…

The first-order part of the Ramsey's Theorem for pairs with an arbitrary number of colors is known to be precisely BSigma03. We compare this to the known division of Ramsey's Theorem for pairs into the weaker principles, EM (the…

Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far reaching generalization of K\"onig's theorem is only known to be true for $r\leq 3$,…

Consider the Hales-Jewett theorem. The $k$-dimensional version of it tells us that the combinatorial space $\mathcal{U}_{M, \Lambda} = \{ \eta \mid \eta: M \to \Lambda \}$ has, under suitable assumptions, monochromatic $k$-dimensional…