  We continue to study the rank functions of tropical matrices. In this paper,
we explain how to reduce the computation of ranks for matrices over the
`supertropical semifield' to the standard tropical case. Using a counting
approach, we prove the existence of a $01$-matrix with many ones and without
large all-one submatrices, and we put our results together and construct an
$n\times n$ matrix with tropical rank $o(n^{0.5+\varepsilon})$ and Kapranov
rank $n-o(n)$.
