TITLE: How close can one get to the missing finite projective planes?
QUESTION [38 upvotes]: 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.

REPLY [16 votes]: I tried this years ago (mid 1990's) and with much slower computers could never get over 290.  Here's one example with 290.
0000000010000000010110000000000010011000000
0000001010010000000000000010000001000001001
1000000001000100000000001000010000000101000
0100000101000000000000010100000000010000001
0010000000000101100000000001000000010000000
1000100110000000101000000000000000100000000
1000011000000001000101010000000000000000000
1001000000000000000010000111000100000000000
0010100000000000000001000100000000001001000
0000000000100000000010010000100000100001000
1010000000001000010000000000101000000000001
0000000001100000100000100010001000001000000
1000000000010010000000100000000000010010000
0000000010100011000000001100000000000000010
0101010010000100000000100000100000000000000
0000000100010001000000000000100100001100000
0000000010000000000000010001001000000100100
0000100000000001000010100000010000000000101
0000000000000000100100000100110001000010000
0000000000100100001100000000000100000000001
0000001000000000000000000000011100110000010
0001000000000000100001000000000010000100011
0001000001000001010000000000000001100000000
0000000000000000001001001010100000010000100
0000010000000000000000001001000000101010001
0000100000000100010000010010000000000010010
0000000011001000000001000000000100000010000
0000101001000010000000000001100010000000000
0100000000000001001000000000001010000011000
0011001100100000000000000000000000000010100
0000000100000110000011000000001001000000000
0100001000001000100010001000000000000000000
0100000000110000010001000001010000000000000
0000000000011100000000000100000010100000100
0000000100001000000100100001000000000001010
0001000000001010001000010000010000001000000
0000110000101000000000000000000001010100000
0110000000000010000100000010000000100100000
0000010000000010110000000000000100000001100
0010000000000000000000111000000111000000000
1100000000000000000000000000000001001000110
0000001000000000011000100100000000000100000
0010010001010000001010000000000000000000010