Largest exponent for free-set size in planar graphs

Determine the supremum exponent α such that every n-vertex planar graph contains a free set—i.e., a vertex subset S that can be mapped to any set P of |S| points in the plane while admitting a straight-line crossing-free drawing—of size Ω(n^α). Ascertain whether α equals the shortness exponent σ for cubic triconnected planar graphs by proving or refuting that every triangulation G has a free set of size Ω(c(G^*)), where c(G^*) is the circumference of the dual G^*.

Background

This survey connects the size of largest free sets in triangulations to the circumference of their dual cubic triconnected planar graphs. Known upper bounds on dual circumference (shortness exponents) yield upper bounds on free-set size, while various constructive methods yield lower bounds in subclasses.

Currently, the best general bounds imply 1/2 ≤ α ≤ 0.9859, with the upper bound coming from Grünbaum–Walther’s shortness exponent for cubic triconnected planar graphs. Establishing the exact α would unify the combinatorial and geometric aspects of free sets with extremal cycle-length behavior in duals.