# A note on Hindman-type theorems for uncountable cardinals

## TL;DR

This paper explores Hindman-type theorems for uncountable cardinals, adapting recent countable results to establish new variants and bounds for these theorems in uncountable settings.

## Contribution

It introduces natural Hindman-type theorems for uncountable cardinals by adapting recent countable results and provides lower bounds for some variants.

## Key findings

- Established new Hindman-type theorems for uncountable cardinals.
- Demonstrated how to adapt countable results to uncountable settings.
- Provided lower bounds for certain Hindman-type variants.

## Abstract

Recent results of Hindman, Leader and Strauss and of Fern\'andez-Bret\'on and Rinot showed that natural versions of Hindman's Theorem fail {\em for all} uncontable cardinals. On the other hand, Komj\'ath proved a result in the positive direction, showing that {\em there are} arbitrarily large abelian groups satisfying {\em some} Hindman-type property. In this note we show how a family of natural Hindman-type theorems for uncountable cardinals can be obtained by adapting some recent results of the author from their original countable setting. We also show how lower bounds for {\em some} of the variants considered can be obtained.