# A parametrized diamond principle and union ultrafilters

## TL;DR

This paper explores a new cardinal invariant linked to Hindman's theorem, demonstrating its small size in a specific forcing model and showing that a related diamond principle guarantees the existence of union ultrafilters, including in the iterated Sacks model.

## Contribution

It introduces a parametrized diamond principle that implies the existence of union ultrafilters and proves their existence in the iterated Sacks model of Set Theory.

## Key findings

- The cardinal invariant is small in the iterated Sacks perfect set forcing model.
- The parametrized diamond principle implies the existence of union ultrafilters.
- Union ultrafilters exist in the iterated Sacks model.

## Abstract

We consider a cardinal invariant closely related to Hindman's theorem. We prove that this cardinal invariant is small in the iterated Sacks perfect set forcing model, and that its corresponding parametrized diamond principle implies the existence of union ultrafilters. As a corollary, this establishes the existence of union ultrafilters in the iterated Sacks model of Set Theory.