Question 237, Let $S=\{1,2, \ldots, n\}, A_{i}=\left\{a_{i}, b_{i}, c_{i}\right\}\left(a_{i}<b_{i}<c_{i}, i=1, 2, \ldots, m\right)$ be a three-element subset of $S$, for any $A_{i}, A_{j}(1 \leq i<j \leq m), a_{i}=a_{j}, b_{i}=b_{j}, c_{i}=c_{j}$ at most one of these can be true, find the maximum value of $m$.

For any $A_{1}=\left\{a_{i}, b_{i}, c_{i}\right\}, b_{i} \in\{2,3, \ldots, n-1\}$. And when $b_{i}=k \in\{2,3, \ldots, n-1\}$, $a_{i} \in\{1,2, \ldots, k-1\}, b_{i} \in\{k+1, k+2, \ldots, n\}$. This indicates: the number of $A_{i}$ satisfying $\mathrm{b}_{1}=k$ does not exceed $\min \{k-1, n-k\}$. Therefore, we have:
$$
\mathrm{m} \leq \sum_{\mathrm{k}=2}^{\mathrm{n}} \min \{\mathrm{k}-1, \mathrm{n}-\mathrm{k}\}=\left[\frac{(\mathrm{n}-1)^{2}}{4}\right]
$$

On the other hand, take all the ternary subsets $\{a, b, c\}$ that satisfy $a+c=2 b(1 \leq a<b<c \leq n, a, b, c \in Z)$. There are exactly $\left[\frac{(\mathrm{n}-1)^{2}}{4}\right]$ such ternary subsets, and these $\left[\frac{(\mathrm{n}-1)^{2}}{4}\right]$ ternary subsets meet the conditions of the problem.