For example, $2 X=\{1,2,3, \cdots, n\}, A, B, C$ are partitions of $X$, i.e., $A \cup B \cup C=X$, and the intersections of $A, B, C$ are pairwise empty. If one element is taken from each of $A, B, C$, then the sum of any two is not equal to the third. Find $\max \min (|A|,|B|,|C|)$.
This problem is a bit more challenging than the previous one, but we should still start with specific examples. Let's still consider the parity. If \( A \) consists of the odd numbers in \( X \), and \( B \cup C \) consists of the even numbers in \( X \), then they meet the requirements of the problem. In this case,
\[
\min (|A|,|B|,|C|)=\left[\frac{n}{4}\right] .
\]

From this, we can make the conjecture
\[
\max \min (|A|,|B|,|C|)=\left[\frac{n}{4}\right],
\]
which means that
\[
\min (|A|,|B|,|C|) \leqslant \frac{n}{4} .
\]