4. Let $S$ be a set of $n$ elements. Determine the largest number $m$ for which there exists a family $\left\{S_{1}, S_{2}, \ldots, S_{m}\right\}$ of distinct non-empty subsets of the set $S$ such that the intersection of any three sets from this family is the empty set.
Solution. Let $\left\{S_{1}, S_{2}, \ldots, S_{m}\right\}$ be a family of distinct non-empty subsets of the set $S=\{1,2, \ldots, n\}$ such that the intersection of any three of its elements is an empty set. This means that each element of the set $S$ will be an element of at most two of the sets $S_{1}, S_{2}, \ldots, S_{m}$. If we denote by $\left|S_{i}\right|$ the number of elements of the set $S_{i}$, we get that $\sum_{i=1}^{m}\left|S_{i}\right| \leq 2 n$.

On the other hand, if we denote by $k$ the number of one-element subsets $S_{i}$ that are members of the given family, then $\sum_{i=1}^{m}\left|S_{i}\right| \geq k+2(m-k)$. From the last two inequalities we get $2 m \leq 2 n+k$. But, $k \leq n$, so $2 m \leq 3 n$, or $m \leq\left[\frac{3 n}{2}\right]$.

The following example

$$
\{1\},\{2\}, \ldots,\{n\},\{1,2\},\{3,4\}, \ldots,\left\{2\left[\frac{n}{2}\right]-1,2\left[\frac{n}{2}\right]\right\}
$$

shows that the desired maximum value for $m$ is $\left[\frac{3 n}{2}\right]$.