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A game‐theoretic proof of Shelah's theorem on labeled trees
Author(s) -
Wilson Trevor M.
Publication year - 2020
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201900060
Subject(s) - mathematics , cardinality (data modeling) , homomorphism , partition (number theory) , combinatorics , discrete mathematics , tree (set theory) , data mining , computer science
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.