Question 228, Set $S=\{1,2, \ldots, 10\}$ has several five-element subsets satisfying: any two elements in $S$ appear together in at most two five-element subsets. Ask: What is the maximum number of five-element subsets?
Question 228, Solution: Each two elements form a "binary pair".
We first consider element 1. The "binary pairs" containing 1 are 9 in number, each can appear in at most 2 five-element subsets, so the "binary pairs" containing 1 appear in all five-element subsets at most 18 times. On the other hand, each five-element subset containing element 1 has exactly 4 "binary pairs" containing 1, so the number of five-element subsets containing element 1 does not exceed $\left[\frac{18}{4}\right]=4$. This means that, in these five-element subsets, element 1 appears at most 4 times. Similarly, elements $2, 3, \ldots, 10$ also appear at most 4 times, so the total number of elements in all five-element subsets does not exceed $10 \times 4=40$. Since each five-element subset has 5 elements, the number of five-element subsets does not exceed $\frac{40}{5}=8$.

Below are 8 five-element subsets that meet the conditions:
$$
\begin{array}{l}
\{1,2,3,4,5\},\{1,2,3,6,7\},\{1,4,5,8,9\},\{1,6,7,8,9\}, \\
\{2,4,6,8,10\},\{2,5,7,8,10\},\{3,4,6,9,10\},\{3,5,7,9,10\}
\end{array}
$$

In conclusion, there can be at most 8 five-element subsets.