4 Let $X=\{00,01, \cdots, 98,99\}$ be the set of 100 two-digit numbers, and $A$ be a subset of $X$ such that: in any infinite sequence of digits from 0 to 9, there are two adjacent digits that form a two-digit number in $A$. Find the minimum value of $|A|$. (52nd Moscow Mathematical Olympiad)
4. Let $A_{ij}=\{\overline{ij}, \overline{ji}\}, i, j \in\{0,1,2, \cdots, 9\}$, then $A$ must contain at least one element from $A_{ij}$, otherwise, the infinite sequence ijijijij $\cdots$ would have no adjacent digits belonging to $A$, leading to a contradiction. Clearly, the set $A_{ij}$ has a total of $10+\mathrm{C}_{10}^{2}=55$ elements (where $A_{00}, A_{11}, \cdots, A_{99}$ have 10 elements), so $|A| \geqslant 55$. Furthermore, let $A=\{\overline{ij} \mid 0 \leqslant i \leqslant j \leqslant 9\}$, (i.e., $A_{ij}$ where all indices satisfy $i \leqslant j$), then $|A|=55$. In this case, for any infinite sequence, let its smallest digit be $i$, and the digit following $i$ be $j$, then $i \leqslant j$, so $\overline{ij} \in A$. Therefore, the minimum value of $|A|$ is 55.