3. Let $X$ be a 100-element set. Find the smallest positive integer $n$ such that for any sequence $A_{1}, A_{2}, \cdots, A_{n}$ of subsets of $X$, there exist $1 \leqslant i<j<k \leqslant n$ satisfying $A_{i} \subseteq A_{j} \subseteq A_{k}$ or $A_{i} \supseteq A_{j} \supseteq A_{k}$.
3. $n=\mathrm{C}_{102}^{51}+1$.

Consider the following sequence of subsets: $A_{1}, A_{2}, \cdots, A_{N}$, where $N=\mathrm{C}_{100}^{50}+\mathrm{C}_{100}^{49}+\mathrm{C}_{100}^{51}+\mathrm{C}_{100}^{50}=\mathrm{C}_{102}^{51}$.
The first segment of $\mathrm{C}_{100}^{50}$ terms consists of all 50-element subsets, the second segment of $\mathrm{C}_{100}^{49}$ terms consists of all 49-element subsets, the third segment of $\mathrm{C}_{100}^{51}$ terms consists of all 51-element subsets, and the fourth segment of $\mathrm{C}_{100}^{50}$ terms consists of all 50-element subsets. Since the sets within the same segment do not contain each other, we only need to consider three subsets taken from different segments. It is easy to see that these three sets $A_{i}, A_{j}, A_{k}$ do not satisfy the given condition.
Therefore, the required $n \geqslant \mathrm{C}_{102}^{51}+1$.
Next, we prove that if the sequence of subsets $A_{1}, A_{2}, \cdots, A_{m}$ does not contain $A_{i}, A_{j}, A_{k} (i < j < k)$ such that $A_{i} \subseteq A_{j} \subseteq A_{k}$.
If both are negative, then let $B_{j}=A_{j}$;
If the former is positive and the latter is negative, then let $B_{j}=A_{j} \cup\{x\}$;
If the former is negative and the latter is positive, then let $B_{j}=A_{j} \cup\{y\}$;
If both are positive, then let $B_{j}=A_{j} \cup\{x, y\}$.
We now verify that $B_{1}, B_{2}, \cdots, B_{m}$ are mutually non-contained.
Assume $i < j < k$ such that $A_{j} \subseteq A_{k}$, which leads to $A_{i} \subseteq A_{j} \subseteq A_{k}$, contradicting the assumption.
Similarly, $B_{i} \supseteq B_{j}$ is also impossible.
Thus, $B_{1}, B_{2}, \cdots, B_{m}$ are mutually non-contained subsets of the 102-element set $X \cup\{x, y\}$. By Sperner's theorem, we have $m \leqslant \mathrm{C}_{102}^{51}$.