Suppose $A \in \mathbb{R}^{n \times n}$ has positive entries on it main diagonal and $\mbox{rank}(A) =: d < n$. Then, what is the maximum number of many negative entries $A$ can contain?

If, in addition, $A$ is positive semidefinite (PSD), then using the fact that in $d$-dimensional space there are at most $d+1$ vectors with pairwise negative inner product and Turán's Theorem from graph theory we can bound the number of negative entries by roughly $$\left(\frac{d-1}{d}\right) n^2$$

But what if $A$ is not PSD?

I've been stuck with this problem for a while. Any helpful reference is greatly appreciated!