This question can be interpreted as an instance of the Zarankiewicz problem. Suppose we have an $n\times n$ matrix with entries in $\{0,1\}$ with no $\begin{pmatrix}1 & 1\\ 1& 1\end{pmatrix} $ minor. The problem asks for the maximum possible number of entries equal to $1$. When $n=q^2+q+1$, one may take the incidence matrix of (points vs. lines in) the finite projective plane of order $q$, giving the answer $(q^2+q+1)(q+1)$. Moreover one can prove that this answer is optimal, when a projective plane of the right order exists. Since there is no finite projective plane of order $6$ one may ask

What is the maximum possible number of entries equal to $1$ in such a $43\times 43$ matrix?

The upper bound of $(6^2+6+1)(6+1)=301$ can not be achieved, but is the answer close to it?

The motivation here, is to understand if projective planes are "badly approximable" when they do not exist for a given order. One may speculate that the answer to the question above is given by cutting off from a projective plane of order $7$ a carefully chosen set of $13$ points and lines, but I'm not sure.