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Abstract

Silver's fundamental dichotomy in the classical theory of Borel reducibility states that any Borel (or even co-analytic) equivalence relation with uncountably many classes has a perfect set of classes. The natural generalisation of this to the generalised Baire space $\kappa ^\kappa $ for a regular uncountable $\kappa $ fails in Gödel's $L$, even for $\kappa $-Borel equivalence relations. We show here that Silver's dichotomy for $\kappa $-Borel equivalence relations in $\kappa ^\kappa $ for uncountable regular $\kappa $ is however consistent (with GCH), assuming the existence of $0^\#$.