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

We give a new proof of a theorem of Shelah which states that for every family of labeled trees, if the cardinality κ of the family is much larger (in the sense of large cardinals) than the cardinality λ of the set of labels, more precisely if the partition relation κ → ( ω ) λ < ωholds, 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.