Maximum constant β for one-bend free sets in planar graphs

Determine the largest constant β such that every n-vertex planar graph admits a 1-bend free set—i.e., a vertex subset S that can be mapped to any set P of |S| points while admitting a crossing-free drawing with at most one bend per edge—of size βn−o(n).

Background

One-bend free sets relax straight-line drawings by allowing one bend per edge. The current best general lower bound guarantees size at least 11n/21 via constructions using spanning trees with many leaves.

Determining the optimal constant β would quantify the trade-off between geometric flexibility (allowing one bend) and the size of vertex subsets that can be fixed arbitrarily.