Let $\mathscr{M}[n]$ be collection of $n\times n$ matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is minimum real rank of matrices in $\mathscr{M}[n]$?
Given real rank $r$, then $\mathscr{N}[r]$ be collection of square matrices with real entries from $\{0,1\}$ such that every row is distinct and every column is distinct.
What is largest size of matrix in $\mathscr{N}[r]$?
Is there an algebraic or geometric way to describe these sets?
I know a separation of rank $r$ and size $2^{r/2}\times 2^{r/2}$. Is this the largest?
$M$ be $2^r\times r$ matrix that binary expansion of all integers from $0$ to $2^r-1$.
$J$ be $2^r\times 2^r$ matrix of $1$s.
$0$ be appropriate size matrix of $0$s.
First block row $[0\quad J\quad M]$, second block row $[J\quad 0\quad M]$, third block row $[M'\quad M'\quad 0]$.
Rank is at most $2+2r$ (sum of ranks of blocks).
Instead of $r/2$, can we replace by $r/c$ where $c>1$ is arbitrarily close to $1$ as $r\rightarrow\infty$ or is there a $c$ that is bounded away from $1$?