Maximum 2-covering number of rank-3 uniform-minor-free matroids

Determine the maximum 2-covering number of a rank-3 matroid with no U_{3,t}-restriction.

Background

The paper defines the 2-covering number as the minimum number of rank-at-most-2 sets, or lines, needed to cover the ground set of a matroid. Structural theorems established earlier imply that rank-3 matroids with no U_{3,t}-restriction have 2-covering number O(t^2), while constructions attaining the paper’s lower bounds have 2-covering number Omega(t).

The problem asks for the sharp extremal value. The authors also mention variants for representable matroids and for higher-rank matroids with no U_{3,t}-minor, but the explicitly stated core problem concerns rank 3 and restrictions.