Let $M$ be a $2^n \times 2^n$ matrix over real number field, where the rows and columns are indexed by subsets of $[n] := \{1,2,\ldots,n\}$, and defined as follows,
$
M\_{A, B} = 1
$
if $A \subseteq B$;
$M\_{A, B} = -1$
if $B \subsetneq A$;
$M\_{A, B}$ can take arbitrary value over $\mathbb{R}$.


In words, $M$ is a matrix with some missing entries. Is there some result lower bounds the rank of matrix $M$, for example, $rk(M) \ge n^{\Omega(\log n)}$.



Rank of a matrix with missing entries
Amazingly (to me) the rank can be as low as $n$. Simply define $M\_{A,B}=1$ when $|A| \le |B|$ and $M\_{A,B}=-1$ when $|A| \gt |B|.$ This is consistent with the previous requirements and makes the $A$ row depend only on $|A|.$


