Universal point subsets larger than n^{1/2+ε} for planar graphs

Ascertain whether, for some ε>0, every set of size Ω(n^{1/2+ε}) is a universal point subset for the class of n-vertex planar graphs; equivalently, develop constructions guaranteeing universal point subsets strictly exceeding n^{1/2} by a polynomial factor.

Background

Universal point subsets generalize universal point sets by requiring only |P| vertices of a graph to be placed at the given points. Free-set techniques currently guarantee universal point subsets of size Θ(√n), and there exists no obstruction to linear-size subsets by existing arguments.

Improving beyond √n would sharpen our understanding of partial embedding flexibility and might suggest paths toward the full universal point set problem.