Solution:

There is an assignment of the integers $1,2,3, \ldots, 9$ to the vertices of the regular nonagon that gives $n=16$. Let $S$ be the sum of all sums of the integers assigned to three consecutive vertices. If there are integers assigned to three consecutive vertices whose sum is at most $14$, then $S<135$, which is a contradiction. Thus, every sum of the integers assigned to three consecutive vertices is equal to $15$. Consider $a, b, c, d$. Then $a+b+c=15$ and $b+c+d=15$, which implies that $a=d$, a contradiction again.