A Polychromatic Ramsey Theory for Ordinals

Some results in polychromatic Ramsey theory

Classical Ramsey theory (at least in its simplest form) is concerned with problems of the following kind: given a set X and a colouring of the set [X] of unordered n-tuples from X, find a subset Y ⊆ X such that all n-tuples in [Y ] get the same colour. Subsets with this property are called monochromatic or homogeneous, and a typical positive result in Ramsey theory has the form that when X is l...

More results in polychromatic Ramsey theory

We study polychromatic Ramsey theory with a focus on colourings of [ω2]. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω2 and ω2 →poly (α)2א0−bdd for every α < ω2. (2) 2ω = ω2 and ω2 9poly (ω1)2−bdd

Ramsey Theory of Cardinals, Ordinals, Trees, and Partial Orders

More on Partitioning Triples of Countable Ordinals

Consider an arbitrary partition of the triples of all countable ordinals into two classes. We show that either for each finite ordinal m the first class of the partition contains all triples from a set of type ω +m, or for each finite ordinal n the second class of the partition contains all triples of an nelement set. That is, we prove that ω1 → (ω + m,n)3 for each pair of finite ordinals m and...

Analytic combinatorics, proof-theoretic ordinals, and phase transitions for independence results

This paper is intended to give for a general mathematical audience (including non logicians) a survey about intriguing connections between analytic combinatorics and logic. We define the ordinals below ε0 in non-logical terms and we survey a selection of recent results about the analytic combinatorics of these ordinals. Using a versatile and flexible (logarithmic) compression technique we give ...