Rank of a (0,1) matrix.

Question

Let $A=\left[a_{ij}\right]$ be an $m \times n$, $m\geq n$ zero-one matrix. Is there any relation between $rank(A)$ and sum of its columns?

Thanks in advance.

Answer 1

well, the rank is clearly bounded by the number of positive terms in the sum of all the columns.

If the number of distinct terms in the sum of all columns is $c$ then we can also guarantee $\text {rank} (M)\geq c-\delta$. Where $\delta$ is $1$ if the sum contains a $0$ term and $0$ otherwise.