This is a presentation about the idempotent elements in semigroups and their relations with Ramsey Theory.
Text of Idempotents in Semigroups
Idempotents in compact semigroups and Ramsey Theory
Complete disorder is impossibleT.S.Motzkin
Piegenhole principleIf m objects are some colored with n colors and m>n then two objects have the same color.
Schurs theorem (1916)
Van der Waerdens Theorem (1927)
Ramseys Theorem (1930,finite version)
Let r,k,l be given integers. Then there is a positive integer n with the following property.If the k-subsets of an n-set are colored with r colors,then there is a monochromatic l-set i.e one all of whose k-sets have the same color.
Ramseys theorem (infinite form)Let X be an infinite set, and k and r positive integers. Suppose that the k-subsets of X are colored with r colors. Then there is an infinite subset Y of X, all of whose k-subsets have the same color.
Folkmans TheoremIf N is finitely colored there exist arbitrarily large finite sets A such that FS(A) is monochromatic.
Hindmans TheoremIf N is finitely colored there exists infinite such that FS(S) is monochromatic.
Theorem of Milliken and Taylor
From now on Compact semigroup=compact hausdorf right topological semigroup
Theorem.Any compact semigroup has idempotent elements.
TheoremIf S is a compact semigroup then S has minimal left ideals.