Creature forcing and topological Ramsey spaces

TL;DR

This paper explores the relationship between creature forcings and topological Ramsey spaces, demonstrating that certain sets of pure candidates contain dense subsets with Ramsey properties and establishing a new product tree Ramsey theorem.

Contribution

It introduces a novel connection between creature forcings and topological Ramsey spaces, including new dense subsets and a variant of the product tree Ramsey theorem.

Findings

Identification of dense subsets with Ramsey properties in creature forcing candidates

Development of a new variant of the product tree Ramsey theorem

Establishment of pigeonhole principles for specific examples

Abstract

This article introduces a line of investigation into connections between creature forcings and topological Ramsey spaces. Three examples of sets of pure candidates for creature forcings are shown to contain dense subsets which are actually topological Ramsey spaces. A new variant of the product tree Ramsey theorem is proved in order to obtain the pigeonhole principles for two of these examples.