For an integer $n$, let $[n] = \{1,\cdots,n\}$. A family $N$ of subsets of $[n]$ , i.e. $N\subset 2^{[d]}$ is said to have Property B(i,j) if the following holds:
For every $I,J\subset [n]$ such that $\vert I\vert =i, \vert J\vert=j$ and $I\cap J=\emptyset$, there is a $K\in N$ such that $I\cap K = \emptyset$ and $J\subset K$. My question is that how small the size of such a family can be? I suspect that the answer is $f=f(i,j,n)$ for some function $f$ that depends on n only polynomially(even be independent of n). I am sure this is a known problem.
Towards proving the upper bound, I apply the usual probabilistic method: pick $f$ random subsets of $[n]$ and put them into $N$. A random subset $R$ of $[n]$ is computed by picking every element of $[n]$ with probability $p$ (to be determined). Note that the number of $I,J\subset [n]$ such that $\vert I\vert =i, \vert J\vert=j$ and $I\cap J=\emptyset$ is $A=$ $n\choose i$ $n-i\choose j$. After some calculation and applying union bound, I get that the probability that $N$ has Property B(i,j) is at least
$1-A\big(1-(1-p)^ip^j\big)^f$
Now for specific values of $p$ and $f$, this should be non-zero. I fail to find such values.
Towards the other end of the spectrum, what is the lower bound of the size of such a family?