TITLE: Number of non-zero entries in $A$ and $AA^T$
QUESTION [6 upvotes]: Let $A$ be a matrix with each entry $0$ or $1$ such that each row contains at most $r$ $1$'s and each row of $AA^T$ contains at most $z$ non-zero entries. Can we somehow bound from above the number of $1$'s on each column of $A$ in terms of $r$ and $z$ only? I get lost in the use of the matrix multiplication calculations.
(The bound need not be tight but should be expressible as a simple function of $r$ and $z$.)
Any help appreciated!

REPLY [3 votes]: Any column with $k$ ones in $A$ corresponds to a $k \times k$-submatrix of $AA^T$ with strictly positive entries, which shows $k \leq z$. Taking $A$ with the first column having $z$ ones and all other entries $0$ shows this is sharp, regardless of $r$ (unless $r=0$ of course.)