Universally Baire sets and definable well-orderings of the reals

Abstract

Let n ≥ 3 be an integer. We show that it is consistent (relative to the consistency of n-2 strong cardinals) that every σ¹_n-set of reals is universally Baire yet there is a (lightface) projective well-ordering of the reals. The proof uses “David’s trick” in the presence of inner models with strong cardinals.