Ramsey theory for hypergroups

Ramsey Theory

Ramsey Theory

Ramsey Theory is the study of inevitable substructures in large (usually discrete) objects. For example, consider colouring the edges of the complete graph Kn with two colours. In 1930, Ramsey [13] proved that if n is large enough, then we can find either a red complete subgraph on k vertices or a blue complete subgraph on ` vertices. We write Rpk, `q for the smallest such n. Another famous exa...

Ramsey Theory

We give a proof to arithmetic Ramsey’s Theorem. In addition, we show the proofs for Schur’s Theorem, the Hales-Jewett Theorem, Van der Waerden’s Theorem and Rado’s Theorem, which are all extensions of the classical Ramsey’s Theorem.

Extremal Ramsey theory for graphs

If G and H are graphs, define the Ramsey number to be the least number p such that if the lines of the complete graph Kp are colored red and blue (say),either the red subgraph contains a copy of G or the blue subgraph contains H. Also set r(G) = r(G,G) ; these are called the diagonal Ramsey numbers. These definitions are taken from Chvátal and Harary [1J ; other terminology will follow Harary [...

Ramsey Theory

Ramsey theory is a branch of mathematics that focuses on the appearance of order in a substructure given a structure of a specific size. This paper will explore some basic definitions of and history behind Ramsey theory, but will focus on a subsection of Ramsey theory known as Ramsey numbers. A discussion of what Ramsey numbers are, some examples of their relevance in real-life scenarios, and a...