$2 \cdot 35$ subset $X$ of the set $\{00,01, \cdots, 98,99\}$ satisfies: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. What is the minimum number of elements that $X$ should contain?
[Solution] For any $i, j \in\{0,1, \cdots, 9\}, X$ should contain $i j$ or $j i$ (otherwise the sequence $i j i j i j \cdots$ does not meet the requirement), there are $10+C_{10}^{2}=55$ such unordered ( $i, j$) pairs,

thus $|X| \geqslant 55$.
On the other hand, if $X=\{i j \mid 0 \leqslant i \leqslant j \leqslant 9\}$, then $X=55$, and for any infinite sequence, let $i$ be the smallest digit it contains, and $j$ be the digit following $i$, then $i j \in X$.
Therefore, $X$ must contain at least 55 elements.