There is set $A$ with cardinality $2^n$. For every $x \in A$ there is $A_x$ - subset of $A$ with cardinality $2^m$, $x \in A_x$. $M=\{A_x|x \in A \}$. Are there $B \subset A$ with cardinality $\ge 2^n/Poly_1(n)$ and $C \subset M$ (some subsets of kind $A_x$) that any $x \in B$ is covered by $\ge 1$ and $\le Poly_2(n)$ sets from $C$? ($Poly_i(n)$ are some polynomials).

UPD: $m$ is not depend from $x$.

I can get positive answer if we know that every $x$ is covered by more then $k$ and less then $2k$ sets: then random $x$ don't belong to random set with probability $\ge 1 - 2^{m-n-1}$, so random $x$ don't belongs to $n\cdot2^{n - m + 1}$ sets with probability $\ge (1 - 2^{m-n-1})^{n\cdot2^{n-m + 1}} \approx e^{-n} < 2^{-n}$. So there are $n\cdot2^{n - m + 1}$ sets that covered every element from $A$. Random $x \in A$ is covered by $2n$ these sets and elements that cavered $> 100n$ sets are less then half of $A$. Removing the we getting that want.