We study Borel reduction of equivalence relations (more generally: quasi-orders) in the generalized Baire and Cantor spaces, and connect it with the study of two diamond-type principles. One of these principles holds for all ineffable sets, and the other fails on all ineffable sets, but, in $L$, holds for all stationary non-ineffable sets.

This is a joint work with Gabriel Fernandes and Miguel Moreno.