Consider the set family $F$ of all $t$-element subsets of $[n]$, for some positive integer $n$. 
* Let $P_1$ be a partition of $F$ into $k$ blocks.
* Let $P_2 \ne P_1$ be another partition of $F$ into $k$ blocks. 

**Questions:** 
1. Is there anything known about how small $k$ needs to be such that there are at least two $t$-element subsets $S_1,S_2 \in F$ such that there exist blocks $B_1 \in P_1$ and $B_2 \in P_2$ with the property that $S_1,S_2 \in B_1 \cap B_2$. 

2. If the general case is open, is there anything known for more specific cases, e.g., when  $t\approx \sqrt{n}$?

3. Are there any related problems that I could look at?