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Chapter 1. Introduction Chapter 2. Banach algebras and their second duals Chapter 3. Semigroups Chapter 4. Semigroup algebras Chapter 5. Stone-Čech compactifications Chapter 6. The semigroup (βS, 2) Chapter 7. Second duals of semigroup algebras Chapter 8. Related spaces and compactifications Chapter 9. Amenability for semigroups Chapter 10. Amenability of semigroup algebras Chapter 11. Amenability and weak amenability for certain Banach algebras Chapter 12. Topological centres Chapter 13. Open problems

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A finite or infinite matrix A with rational entries is called partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic vector x with . Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular.While in the finite case partition regularity is well understood, very little is known in the infinite case. Our aim in this paper is to present some of the natural and appealing open problems in the area.