Considering all the answers so far, I thought I might as well add one with a more topological flavor to it
The existence of an L-space was known to be consistent for years (See The Handbook of Set-Theoretic Topology, Chapter 7, pg. 295). It was only recently that a ZFC construction was given here
Some of the existence proofs for certain types of embeddings, and automorphisms between Boolean algebras have this flair to them, (See "The fourth head of $\beta\mathbb{N}$" by Ilijas Farah, in Open Problems in Topology II. pg 139.)
Certain types of gnarly questions about coverings of $\mathbb{R}$ involving the forward and inverse images of $\aleph_1$ many continuous functions, have had some success with this see this answer