A Ramsey theorem for measurable sets

The strength of Ramsey Theorem for coloring relatively large sets

We characterize the computational content and the proof-theoretic strength of a Ramseytype theorem for bi-colorings of so-called exactly large sets. An exactly large set is a set X ⊂ N such that card(X) = min(X) + 1. The theorem we analyze is as follows. For every infinite subset M of N, for every coloring C of the exactly large subsets of M in two colors, there exists and infinite subset L of ...

Brooks’s Theorem for Measurable Colorings

Throughout, by a graph we mean a simple undirected graph, where the degree of a vertex is its number of neighbors, and a d-coloring is a function assigning each vertex one of d colors so that adjacent vertices are mapped to different colors. This paper examines measurable analogues of Brooks’s Theorem. While a straightforward compactness argument extends Brooks’s Theorem to infinite graphs, suc...

Ramsey theorem for designs

We prove that for any choice of parameters k, t, λ the class of all finite ordered designs with parameters k, t, λ is a Ramsey class.

A stochastic Ramsey theorem

A stochastic extension of Ramsey’s theorem is established. Any Markov chain generates a filtration relative to which one may define a notion of stopping time. A stochastic colouring is any fc-valued (k < oo) colour func­ tion defined on all pairs consisting of a bounded stopping time and a finite partial history of the chain truncated before this stopping time. For any bounded stopping time 6 a...

Measurable Utility and the Measurable Choice Theorem