In the following paper (pages 122-23), Erdős asks if there is a constant $c > 0$ such that every subset $A$ of plane of area more than $c$ contains the vertices of a triangle of unit area.

Is this still open? Has anyone discovered interesting lower bounds for $c$?

As a "motivational" puzzle you can show that if $c = \infty$ then $A$ contains vertices of triangles of all possible areas.