7. (China Western Mathematical Olympiad) We call $A_{1}, A_{2}, \cdots, A_{n}$ a $n$-partition of set $A$ if
(1) $A_{1} \cup A_{2} \cup \cdots \cup A_{n}=A$;
(2) $A_{i} \cap A_{j} \neq \varnothing, 1 \leqslant i<j \leqslant n$.

Find the smallest positive integer $m$, such that for any 14-partition $A_{1}, A_{2}, \cdots, A_{14}$ of $A=\{1,2, \cdots, m\}$, there must exist a set $A_{i}(1<i<14)$, in which there are two elements $a, b$ satisfying $b<a \leqslant \frac{4}{3} b$.
7. Solution (1) If $ma > b$, and $a - b \geqslant 14$.
Thus $b \leqslant a - 141 + \frac{14}{42} = \frac{4}{3}$.
Therefore, the positive integer $m \geqslant 56$.
(2) If $m = 56$, then for any partition of $A$ into $A_{1}, A_{2}, \cdots, A_{14}$, among the numbers $42, 43, \cdots, 56$, there must be two numbers belonging to the same $A_{i}$, such that they satisfy $b \leqslant a \leqslant \frac{4}{3} b$.
In summary, the smallest positive integer value of $m$ is 56.