Solution:

Let $S(n)$ be the set of such subsets. Consider the map from $S(n)$ to $S(n + 1)$ that adds one to the largest element of each $A \in S(n)$. This map is an injection (needs proof but easy) and not a surjection provided that $S(n + 1)$ contains a set whose largest and second largest elements differ by one. For even $n = 2k \geqslant 2$ this is true since we can take $\{k, k + 1\} \in S(n + 1)$ and for odd $n = 2k + 1 \geqslant 5$ this is true since we can take $\{1, k, k + 1\}$. So for $n \geqslant 5$, we must have $f(n) < f(n + 1)$ and there do not exist infinitely many such pairs.