Let's say I have 10 finite sets $X_1, X_2, ..., X_{10}$.
Given the values $|X_i|,$ for $i=1,2,..,10$ and $|X_i \cap X_j|$ for $i\not= j$ and $i,j=1,2,...,10$,
what is the best estimation of the cardinality of
$$|X_1 \cap X_2 \cap ... \cap X_{10}|$$
that I can get (without knowing the elements of the 10 sets $X_i$)?
My guess is that the only thing that I can assume with accuracy is that
$$|X_1 \cap X_2 \cap ... \cap X_{10}| \leq min_{i,j}(|X_i \cap X_j|), \text{ with } i\not=j \text{ and }i,j=1,...,10.$$
Can I do any better?
All the Best,
Luca
You need more information to get an accurate count of the intersection, otherwise all you can say is what you have said
$$|X_1 \cap X_2 \cap ... \cap X_{10}| \leq min_{i,j}(|X_i \cap X_j|), \text{ with } i\not=j \text{ and }i,j=1,...,10.$$
Note that even in the simple case of three sets, we have
$$ |A\cup B\cup C|= |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$$
Thus with the given information and without knowing $$ |A\cup B\cup C|$$ we can not find $$ |A\cap B\cap C|$$