Consider a finite set $X$ and some subsets $X_i \subset X$, $i = 1,\dots,n$, with $\bigcup_i X_i = X$, of which one only knows the sizes $|X_i|$ and $|X_i \cap X_j|$.
How do I systematically construct a set system $\{Y_i\}$ with $|Y_i| = |X_i|$ and $|Y_i \cap Y_j| = |X_i \cap X_j|$?
I would start by creating $n$ sets $Y_i = \Big\{y_i^1,y_i^2,\dots,y_i^{|X_i|}\Big\}$ with $y_i^k \neq y_j^l$ for $i \neq j$.
Then I would pick a pair of sets $Y_i, Y_j$ and choose uniformly at random $n_{ij} = |X_i \cap X_j|$ elements from $Y_i$ and $Y_j$, respectively, getting two $n_{ij}$-element sets $Y_i', Y_j'$.
Then I would identify the elements of $Y_i', Y_j'$ pairwise by a random bijection $\mu_ {ij}: Y_i' \rightarrow Y_j'$. The remaining sets $Y_i \setminus Y_i'$ and $Y_j \setminus Y_j'$ are disjoint and in following steps may not become joint.
Then I would choose another set $Y_k$ and do the same with $Y_j$ and $Y_k$.
Doing the same for $Y_k$ and $Y_i$ might become tricky - or even impossible. Choosing $|X_k \cap X_i|$ elements from $Y_k$ and $Y_i$ at random and identifying them pairwise may not work anymore. So already the step before must be taken with caution.
Which (randomized) algorithm does solve the problem?
Follow-up question: Assume that I managed to construct such a set system $\{Y_i\}$ by some randomized algorithm. What will be the expected size of $Y = \bigcup_i Y_i$?
