4. For the set $\{00,01, \cdots, 98,99\}$, a subset $X$ 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?
4. For any $i, j \in\{0,1, \cdots, 9\}, \lambda$ should include $i j$ or $i i$ (otherwise the sequence $i j i j i \ldots$ would satisfy the problem's requirements). There are $10+C_{10}^{2}=55$ such unordered pairs $(i, j)$, hence $|X| \geqslant 55$.

On the other hand, if we take $X=\{i j: 0 \leqslant i \leqslant j \leqslant 9\}$, then $|X|=55$, and for any non-decreasing 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.