# A game-theoretic proof of Shelah's theorem on labeled trees

## TL;DR

This paper introduces a game-theoretic proof of Shelah's theorem on the existence of homomorphisms between labeled trees under certain large cardinal conditions, providing a novel perspective on the theorem.

## Contribution

The paper presents the first game-theoretic proof of Shelah's theorem, connecting large cardinal partition relations with homomorphisms in labeled trees.

## Key findings

- Game-theoretic characterization of homomorphisms
- Proof of Shelah's theorem using large cardinal assumptions
- New insights into labeled trees and their mappings

## Abstract

We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality $\kappa$ of the family is much larger (in the sense of large cardinals) than the cardinality $\lambda$ of the set of labels, more precisely if the partition relation $\kappa \to (\omega)^{\mathord{<}\omega}_\lambda$ holds, then there is a homomorphism from one labeled tree in the family to another. Our proof uses a characterization of such homomorphisms in terms of games.