Strong general bound for non-representable matroids

Establish that for every finite set V of dimension vectors, each having at most t coordinates, with k equal to the maximum coordinate appearing in any vector in V, the covering threshold for simple matroids satisfies C_M(V) ≤ binom(t+k+1,k+1).

Background

The paper defines C_M(V) as the least integer N such that a simple matroid on N elements whose restrictions are all covered by V must itself be covered by V. The authors prove the weaker bound C_M(V) ≤ Σ_{i=0}^{k+1} t^i for arbitrary matroids, while their algebraic argument gives the sharper binomial bound for representable matroids.

The conjecture asks whether the sharper representable-matroid bound remains valid for all matroids, including non-representable ones. The authors note that any extremal counterexample or equality case would likely need substantial structure, because matroids with many small circuits tend to be representable.