Von Neumann Conjecture

Conjecture
Let $G$ be a group.

Then $G$ is non-amenable iff $G$ contains a free subgroup on two generators.

Resolution
This conjecture was answered in the negative, by Alexander Ol'shanskii.

He demonstrated that the Tarski monster is a non-amenable group with no such free subgroup.

Also see

 * Tarski Monster Group is Non-Amenable with no Free Subgroup of Two Generators