Ramsey Theory and Extremal Combinatorics

In incidence geometry, the De Bruijn-Erdős theorem, originally published by Nicolaas Govert de Bruijn and Paul Erdős in 1948, states a lower bound on the number of lines determined by n points in a projective plane.

Ramsey. This initiated the combinatorial theory now called Ramsey theory, that seeks regularity amid disorder: general

did not publish it until 1961. It is part of the field of combinatorics, and one of the central results of extremal set

in discrete mathematics, describes the largest possible families of finite sets none of which contain any other sets in the family.

result that makes precise one of the corollaries of Ramsey's theorem. While Ramsey's theorem makes it easy

Van der Waerden's theorem is a theorem in Ramsey theory. Van der Waerden's theorem states that for any

of them occurs. This lemma is most commonly used in the probabilistic method, in particular to give existence proofs. There

theorem is a fundamental combinatorial result of Ramsey theory, named after Alfred W. Hales and Robert I. Jewett

the latter being one of the most famous open problems of extremal combinatorics.

the Erdős-Rado theorem is a basic result extending Ramsey's theorem to uncountable sets. It is named after

The Boolean Pythagorean triples problem is a problem from Ramsey theory about whether the positive integers can be colored

is a mathematical theorem in an area of mathematics called Ramsey theory.

in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more

In mathematics, the probabilistic method is a nonconstructive method, primarily used in