Partition theorems from creatures and idempotent ultrafilters

TL;DR

This paper develops a general framework for Ramsey-type partition theorems for countable sets of finite functions, utilizing creature forcing and idempotent ultrafilters, extending classical combinatorial results like Hindman's theorem.

Contribution

It introduces a new scheme connecting creature forcing with ultrafilter techniques to derive novel partition theorems.

Findings

Established a general scheme for Ramsey-type results using creature forcing.

Proved the existence of idempotent ultrafilters relevant to the framework.

Derived new partition theorems related to creature forcings.

Abstract

We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.