Maximum 2-covering number of rank-3 U_{3,t}-free matroids

Determine the maximum 2-covering number among rank-3 matroids with no U_{3,t}-restriction.

Background

The authors prove structural results showing that rank-3 matroids with no U_{3,t}-restriction can be covered by O(t2) lines, while constructions attaining their lower bounds have 2-covering number of order Omega(t).

The precise extremal growth of this covering parameter is therefore unresolved and may clarify the structure of rank-3 matroids avoiding large uniform restrictions.

References

Find the maximum $2$-covering number for rank-$3$ matroids with no $U_{3,t}$-restriction.

— Turán densities for matroid basis hypergraphs  (2502.03673 - Pol et al., 5 Feb 2025) in Problem in Section 7.2, “Discrete geometry”