$\DeclareMathOperator\null{null}$Let $H$ be a hyperplane of the paving matroid $M$ with $r(M)=n$. How large can $\null(H)$ be?
We know that $\null(H)=|H|-r(H)=|H|-(n-1)$. So everything boils down to finding the size of the largest hyperplane in $M$.



Nontrivial upper bounds for the nullity of hyperplanes in paving matroids
There is no such bound. Take many vectors in a real hyperplane $H$ in $\mathbb{R}^n$ in general position, and one vector outside $H$. They form a paving matroid, since all circuits have size $n-1$ or $n$.


