Ramsey Theory: Milliken’s Tree Theorems

Speciale: Peter Rützou Klinting

Titel:  Ramsey Theory: Milliken’s Tree Theorems

Abstract: In this thesis we study Ramsey theory, in particular some major theorems discovered 1930-1981. Ramsey theory is essentially the theory concerning generalisations of the well-known (finite) Ramsey theorem and, in turn, the classic pigeonhole principle. The goal of this thesis is to present and prove two Ramsey theorems by Keith Milliken concerning Ramsey properties in the setting of finite tuples of infinite-heighted, finite-splitting trees. The 1-dimensional version of the first of these theorems can be stated as: if T is an infinite-heighted tree, h a positive integer, and we assume a finite partition of the collection of h-heighted
strongly embedded subtrees in T, then there exists an infinite-heighted strongly embedded subtree S of T such that all h-
eighted strongly embedded subtrees in S belong to a common block of the partition. The 1-dimensional version of the second of Milliken’s theorems states that if T is an infinite-heighted tree, T is the so-called tree topology on T, and R is a collection of infinite-heighted strongly embedded subtrees of T, then R is completely Ramsey if and only if R has the Baire Property in T. We will also present and prove special cases of these theorems that were major stepping stones for discovering Milliken’s theorems,
namely the Halpern-Läuchli theorem and Ellentuck’s theorem, respectively.

Vejleder:  Asger Dag Törnquist, Henrik Granau Holm
Censor:    David Kyed, SDU