 Fremlin and Miller 
 
proved that Menger's conjecture is false, by showing that there is,
in ZFC, a set of real numbers that is Menger but not σ-compact.The Fremlin-Miller proof was dichotomic, and the set witnessing the failure
of the conjecture heavily depends on whether a certain (undecidable) axiom
holds or not.Bartoszyński and Tsaban
 
gave a uniform ZFC example of a Menger subset of the real line that is not σ-compact.