When is the intersection of $k$ sets non-empty?

Question

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!

Answer 1

Your guess is right. For instance, taking $k=3$ and $S=\{1,2,3\}$, you could take $A_1=\{1,2\}$, $A_2=\{1,3\}$ and $A_3=\{2,3\}$.

To find a condition on $S$ that works, it helps to think about the complements of the $A_i$. Write $B=S\setminus A_i$. Then $$\bigcap_{i=1}^k A_i=S\setminus \bigcup_{i=1}^k B_i.$$ The condition is now obvious: to be sure that $\bigcup_{i=1}^k B_i$ is not all of $S$ using a uniform condition on their sizes, you need to know that $|B_i|<|S|/k$ for all $i$. This means you need to know that $$|A_i|>\frac{k-1}{k}|S|$$ for all $i$. You can see that for $k=2$ this is the same as $|A_i|>|S|/k$, but for larger $k$ it is a much stronger condition.