Say that $m$ is the *additive dimension* of $n\in\Bbb N$, and write $m=\operatorname{ad}n$, if $m$ is the greatest integer for which there is an irredundant $m$-element set $M\subset\Bbb N$ that provides a partition of $n$ uniquely: namely
$$n=k_1+\cdots+k_r\quad\text{with}\quad\{k_i:i=1,...,r\}=M,$$
for some $r\in\Bbb N$, where no other partition of $n$ involves only elements of $M$.

As an illustration, take $n=34$. We can choose $M=\{7,8,12\}$ (for example) in this case, with $34=7+7+8+12$, and so $\operatorname{ad}34=3$, since no other partition of $34$ can be formed from the elements of $M$, while any $4$-element set providing a partition of $34$ has a proper subset that does so too. (The last claim entails some checking.)

Note that $\operatorname{ad}0=0$; $\operatorname{ad}n=1$ for $n=1,2,3,4,6$; $\operatorname{ad}n=2$ for $n=5$ and $n=7,...,16$; and $\operatorname{ad}17=3$ (take $M=\{4,6,7\}$). Thus $17$ is the least integer of additive dimension $3$.

What is the least integer of additive dimension $4$? Further, what is the asymptotic behaviour of $\operatorname{ad}n$ as $n$ becomes large?

(This question was earlier posted on MathStackExchange but was not answered.)