My favorite are the results in computability theory that rely on Martin's cone theorem (if A is sufficiently definable degree invariant set (Certainly if Borel but I think more) then either A or it's complement contain a cone.

This can be used for all sorts of neat results like the existence of a cone of minimal covers (both under Turing and arithmetic reducibility...the former has a constructive proof but I'm unaware of one for the later).

A number of other examples here

My favorite are the results in computability theory that rely on Martin's cone theorem (if A is sufficiently definable degree invariant set (Certainly if Borel but I think more) then either A or it's complement contain a cone.

This can be used for all sorts of neat results like the existence of a cone of minimal covers (both under Turing and arithmetic reducibility...the former has a constructive proof but I'm unaware of one for the later). See Odifreddi volume 2 for the one about the arithmetic degrees and volume 1 for the Turing degree claim.

A number of other examples here

Also see this discussion where Noah answered a q of mine about 2-lubs by hitting it over head with set theoretic forcing.