We show a dichotomy for analytic equivalence relations, implying prime relations for certain homomorphisms.
We prove the following dichotomy. Given an analytic equivalence relation [Formula: see text], either [Formula: see text] or else any Borel homomorphism from [Formula: see text] to [Formula: see text] is “very far from a reduction”, specifically, it factors, on a comeager set, through the projection map [Formula: see text] for some [Formula: see text]. As a corollary, we prove that [Formula: see text] is a prime equivalence relation, answering a question on Clemens.
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Assaf Shani (2025) studied this question.