Maximum number of lines covering points with no large general-position set

Determine, among all sets of n vectors in the real vector space R^{s-1} containing no t vectors in general position, the maximum number of lines required to cover the vectors, with t>=s>=3.

Background

The paper relates extremal basis-counting problems for affine matroids to discrete geometry. In rank three, the corresponding question asks how many lines are needed to cover a set of planar points while forbidding t points in general position.

This problem is presented as a geometric counterpart to the paper’s rank-3 matroid questions and is motivated by connections with extremal problems on points and lines.

References

Among all sets of $n$ vectors in ${s-1}$ with no $t$ in general position, what is the maximum number of $s$-sets in general position?

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