Sharp bounds for the midrange crossing constant

Improve the known lower and upper bounds for the midrange crossing constant and determine whether the upper-bound constructions of Pach and Toth and of Czabarka et al. are asymptotically optimal up to lower-order terms.

Background

The midrange crossing constant c is known only within the interval 0.036 ≤ c ≤ 0.09. The lower bound comes from the best known improvements to the Crossing Lemma, while the upper bounds arise from explicit graph constructions based on geometric and random point configurations.

The unresolved issue is both quantitative—improving either endpoint of the interval—and structural: determining whether the cited constructions achieve the optimal asymptotic crossing density.