Linear-size universal point sets for planar graphs

Determine whether there exists a constant c such that, for every n, there is a point set of size c·n that is universal for all n-vertex planar graphs (i.e., supports straight-line crossing-free embeddings of every n-vertex planar graph).

Background

Universal point sets are central in planar graph drawing. It is known that no universal point set of size n exists for all n-vertex planar graphs, while O(n^2) point sets suffice. The current best lower bound is 1.239n−o(n), and the best upper bound is n^2/4−O(n).

A linear-size universal point set would significantly advance geometric graph theory and bridge the gap between known lower and upper bounds.