Suppose that we have a universe $U= \{1,\dots,n\}$ for some integer $n$. How many subsets $S_1,S_2,\dots\subseteq U $ are there such that $S_i \nsubseteq S_j$ but $S_i \cap S_j \neq \emptyset$ for all $i,j$? That means all $S_i$ and $S_j$ pairwise intersect but none are subsets of each other.
I tried starting with one $S_1$ and then counting the number of possible $S_2$ sets using the binomial coefficient but this didn't get me anywhere.