Idempotents in compact Idempotents in compact semigroups and Ramsey semigroups and Ramsey

TheoryTheory

Complete disorder is Complete disorder is impossibleimpossible

T.S.MotzkinT.S.Motzkin

Piegenhole principlePiegenhole principle

If m objects are some colored with n colors and If m objects are some colored with n colors and m>n then two objects have the same color.m>n then two objects have the same color.

Schur’s theorem (1916)Schur’s theorem (1916)

zy that xso ,color same thehaving

N, zy,exist x, thereN of coloring-r allFor

Van der Waerden’s Theorem (1927)Van der Waerden’s Theorem (1927)

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nprogressio arithmetic ticmonochroma a

exists therecolored-r is N if ,r l, all

For

Ramsey’s Theorem (Ramsey’s Theorem (1930,finite 1930,finite versionversion))

Let r,k,l be given integers. Then there is a Let r,k,l be given integers. Then there is a positive integer n with the following property.If positive integer n with the following property.If the k-subsets of an n-set are colored with r the k-subsets of an n-set are colored with r colors,then there is a monochromatic l-set i.e colors,then there is a monochromatic l-set i.e one all of whose k-sets have the same color.one all of whose k-sets have the same color.

Ramsey’s theorem (Ramsey’s theorem (infinite forminfinite form))

Let X be an infinite set, and k and r positive Let X be an infinite set, and k and r positive integers. Suppose that the k-subsets of X are integers. Suppose that the k-subsets of X are colored with r colors. Then there is an infinite colored with r colors. Then there is an infinite subset Y of X, all of whose k-subsets have the subset Y of X, all of whose k-subsets have the same color.same color.

Definition .Definition .

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; Let

sss

Sss

NA

Folkman’s TheoremFolkman’s Theorem

If N is finitely colored there exist If N is finitely colored there exist arbitrarily large finite sets A such that arbitrarily large finite sets A such that FS(A) is monochromatic.FS(A) is monochromatic.

Hindman’s TheoremHindman’s Theorem

If N is finitely colored there exists If N is finitely colored there exists infinite such that FS(S) is infinite such that FS(S) is monochromatic.monochromatic.

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