# Category: Ramsey theory

Piecewise syndetic set
In mathematics, piecewise syndeticity is a notion of largeness of subsets of the natural numbers. A set is called piecewise syndetic if there exists a finite subset G of such that for every finite sub
The Mathematical Coloring Book
The Mathematical Coloring Book: Mathematics of Coloring and the Colorful Life of Its Creators is a book on graph coloring, Ramsey theory, and the history of development of these areas, concentrating i
Green–Tao theorem
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for e
Ramsey's theorem
In combinatorics, Ramsey's theorem, in one of its graph-theoretic forms, states that one will find monochromatic cliques in any edge labelling (with colours) of a sufficiently large complete graph. To
Schur's theorem
In discrete mathematics, Schur's theorem is any of several theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of Axel Schur. In functional analysis, Schu
Rado's theorem is a theorem from the branch of mathematics known as Ramsey theory. It is named for the German mathematician Richard Rado. It was proved in his thesis, Studien zur Kombinatorik.
Erdős–Szekeres theorem
In mathematics, the Erdős–Szekeres theorem asserts that, given r, s, any sequence of distinct real numbers with length at least (r − 1)(s − 1) + 1 contains a monotonically increasing subsequence of le
Sim (pencil game)
Sim is a pencil-and-paper game that is played by two players.
Graham's number
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory. It is much larger than many other large numbers such as Skewes'
Theorem on friends and strangers
The theorem on friends and strangers is a mathematical theorem in an area of mathematics called Ramsey theory.
Milliken's tree theorem
In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets. Let T be a finitely splitting ro
Happy ending problem
In mathematics, the "happy ending problem" (so named by Paul Erdős because it led to the marriage of George Szekeres and Esther Klein) is the following statement: Theorem — any set of five points in t
Slicing the Truth
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of Combinatorial Principles is a book on reverse mathematics in combinatorics, the study of the axioms needed to pro
Partition regularity
In combinatorics, a branch of mathematics, partition regularity is one notion of largeness for a collection of sets. Given a set , a collection of subsets is called partition regular if every set A in
Milliken–Taylor theorem
In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor. Let denote the set
IP set
In mathematics, an IP set is a set of natural numbers which contains all finite sums of some infinite set. The finite sums of a set D of natural numbers are all those numbers that can be obtained by a
Large set (Ramsey theory)
In Ramsey theory, a set S of natural numbers is considered to be a large set if and only if Van der Waerden's theorem can be generalized to assert the existence of arithmetic progressions with common
Erdős–Dushnik–Miller theorem
In the mathematical theory of infinite graphs, the Erdős–Dushnik–Miller theorem is a form of Ramsey's theorem stating that every infinite graph contains either a countably infinite independent set, or
Ramsey class
In the area of mathematics known as Ramsey theory, a Ramsey class is one which satisfies a generalization of Ramsey's theorem. Suppose , and are structures and is a positive integer. We denote by the
Graham–Rothschild theorem
In mathematics, the Graham–Rothschild theorem is a theorem that applies Ramsey theory to combinatorics on words and combinatorial cubes. It is named after Ronald Graham and Bruce Lee Rothschild, who p
Corners theorem
In arithmetic combinatorics, the corners theorem states that for every , for large enough , any set of at least points in the grid contains a corner, i.e., a triple of points of the form with . It was
Folkman's theorem
Folkman's theorem is a theorem in mathematics, and more particularly in arithmetic combinatorics and Ramsey theory. According to this theorem, whenever the natural numbers are partitioned into finitel
Cap set
In affine geometry, a cap set is a subset of (an -dimensional affine space over a three-element field) with no three elements in a line.The cap set problem is the problem of finding the size of the la
Burr–Erdős conjecture
In mathematics, the Burr–Erdős conjecture was a problem concerning the Ramsey number of sparse graphs. The conjecture is named after Stefan Burr and Paul Erdős, and is one of many conjectures named af
Ergodic Ramsey theory
Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory.
Pigeonhole principle
In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. For example, if one has three gloves
Structural Ramsey theory
In mathematics, structural Ramsey theory is a categorical generalisation of Ramsey theory, rooted in the idea that many important results of Ramsey theory have "similar" logical structure. The key obs
Hales–Jewett theorem
In mathematics, the Hales–Jewett theorem is a fundamental combinatorial result of Ramsey theory named after Alfred W. Hales and Robert I. Jewett, concerning the degree to which high-dimensional object
Boolean Pythagorean triples problem
The Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored red and blue so that no Pythagorean triples consist of all red or all blue me
Szemerédi's theorem
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured that every set of integers A with posit
Gowers' theorem
In mathematics, Gowers' theorem, also known as Gowers' Ramsey theorem and Gowers' FINk theorem, is a theorem in Ramsey theory and combinatorics. It is a Ramsey-theoretic result about functions with fi
Ramsey theory
Ramsey theory, named after the British mathematician and philosopher Frank P. Ramsey, is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a known
Erdős–Hajnal conjecture
In graph theory, a branch of mathematics, the Erdős–Hajnal conjecture states that families of graphs defined by forbidden induced subgraphs have either large cliques or large independent sets. It is n
Halpern–Läuchli theorem
In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theo
Van der Waerden's theorem
Van der Waerden's theorem is a theorem in the branch of mathematics called Ramsey theory. Van der Waerden's theorem states that for any given positive integers r and k, there is some number N such tha
Van der Waerden number
Van der Waerden's theorem states that for any positive integers r and k there exists a positive integer N such that if the integers {1, 2, ..., N} are colored, each with one of r different colors, the