Example 5 Let $n(\geqslant 3)$ be a positive integer, and $M$ be an $n$-element set. Find the maximum positive integer $k$ such that there exists a family $\psi$ of $k$ three-element subsets of $M$ with the property that the intersection of any two elements (note: elements of $\psi$ are three-element sets) is non-empty.
When $n \leqslant 5$, it is obvious that the maximum positive integer $k=C_{n}^{3}$.
Now we prove that when $n \geqslant 6$, the maximum positive integer $k=C_{n-1}^{2}$.
On the one hand, let $M=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$, then the family of all three-element subsets of $M$ containing $a_{1}$ satisfies the condition, hence, the positive integer $k \geqslant C_{n-1}^{2}$.
On the other hand, let the maximum integer sought be $t$.
We consider the circular permutations of $a_{1}, a_{2}, \cdots, a_{n}$, then the total number of circular permutations is $(n-1)!$.
Consider the number of circular permutations in which the three elements of some $A_{i}$ in $\psi$ are adjacent, which is $x$, then $x=3!\cdot(n-3)!$.
Notice that when the three elements of $A_{i}$ are adjacent in the circular permutation, there can be at most two other elements $A_{j}$ and $A_{r}$ in $\psi$ whose elements are also adjacent in this circular permutation.

In fact, if $A_{i}=\left\{a_{i}, a_{i,}, a_{i,}\right\}$, then when the elements of $A_{1}$ and $A_{r}$ are also adjacent, it can only be that $A_{j} \cap A_{i}=\left\{a_{i=}, a_{i,}\right\}, A_{r} \cap A_{i}=\left\{a_{i,}, a_{i}\right\}$, or its symmetric form. Otherwise, $A_{i} \cap A_{r}=\varnothing$, which contradicts the problem's condition.
Therefore, there cannot be another element $A_{m}$ in $\psi$ such that the elements of $A_{m}$ are also adjacent in this circular permutation.
The above discussion shows that $t \cdot x \leqslant 3 \cdot((n-1)!)$,
Thus, $t \leqslant \frac{(n-1)!}{2!(n-3)!}=C_{n-1}^{2}$.
Note: The above problem is a special case of the Erdös-柯召一Rado theorem.
In general, let $X$ be an $n$-element set, $r \in \mathbf{N}^{*}, 2 \leqslant r \leqslant \frac{n}{2}, y=\left\{A_{1}, A_{2}, \cdots, A_{t}\right\}$ be a family of subsets of $X$, such that $\left|A_{i}\right|=r(1 \leqslant i \leqslant t)$, and for $1 \leqslant i<j \leqslant t$, we have $A \cap \cap A_{j} \neq \varnothing$, then the maximum value of $t$ is $C_{n-1}^{r-1}$.