On Boolean matrices with full factor rank

It is demonstrated that every $(0,1)$-matrix of size $n\times m$ having Boolean rank $n$ contains a column with at least $\sqrt{n}/2-1$ zero entries. This bound is shown to be asymptotically optimal. As a corollary, it is established that the size of a full-rank Boolean matrix is bounded from above by a function of its tropical and determinantal ranks.
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