Topological Ramsey numbers and countable ordinals

@article{Caicedo2015TopologicalRN,
title={Topological Ramsey numbers and countable ordinals},
author={Andr{\'e}s Eduardo Caicedo and Jacob Hilton},
journal={arXiv: Logic},
year={2015}
}
• Published 30 September 2015
• Mathematics
• arXiv: Logic
We study the topological version of the partition calculus in the setting of countable ordinals. Let α and β be ordinals and let k be a positive integer. We write β →top (α, k)² to mean that, for every red-blue coloring of the collection of 2-sized subsets of β, there is either a red-homogeneous set homeomorphic to α or a blue-homogeneous set of size k. The least such β is the topological Ramsey number Rtop(α, k). We prove a topological version of the Erdős-Milner theorem, namely that Rtop(α, k…
6 Citations
THE POLARISED PARTITION RELATION FOR ORDER TYPES
• Mathematics
The Quarterly Journal of Mathematics
• 2020
We analyse partitions of products with two ordered factors in two classes where both factors are countable or well-ordered and at least one of them is countable. This relates the partition
Infinite and Finitary Combinatorics Around Hrushovski Constructions
The thesis is composed of two distinct and independent parts. Fräıssé-Hrushovski constructions In the first part of the thesis we investigate relationships between several variations and
On the closed Ramsey numbers $R^{cl}(\omega+n,3)$
• Mathematics
• 2020
In this paper, we contribute to the study of topological partition relations for pairs of countable ordinals and prove that, for all integers $n \geq 3$, \begin{align*} R^{cl}(\omega+n,3) &\geq
Calculating the closed ordinal Ramsey number Rcl(ω · 2, 3)
We show that the closed ordinal Ramsey number Rcl(ω · 2, 3) is equal to ω3 · 2.
$R^{cl}(\omega^2,3) = \omega^6$
Closed ordinal Ramsey numbers are a topological variant of the classical (ordinal) Ramsey numbers. We compute the exact value of the closed ordinal Ramsey number $R^{cl}(\omega^2,3) = \omega^6$.
On the closed Ramsey numbers Rcl(ω + n, 3)
• Israel Journal of Mathematics
• 2021