Suppose, given a ground set $S$, we have two subsets $A,B \subseteq S$.
If we know that $|A|, |B| > \frac{|S|}{2}$, then we know that $A \cap B \neq \emptyset$.
Can this be generalized to $k$ sets? Does it hold that when $|A_1|, \dotsc, |A_k| > \frac{|S|}{k}$ for subsets $A_i \subseteq S$, then $\bigcap_{i=1}^k A_i \neq \emptyset$? My guess is no, that it only hold that there are $i,j$ such that $A_i \cap A_j \neq \emptyset$.
But I would like to have a sufficient condition to conclude that the overall intersection is not empty. Do you know an easy one?
Thanks a lot!