OPEN
Let $k\geq 3$ and $A$ be an additive basis of order $k$. Does there exist a constant $c=c(k)>0$ such that if $r(n)\geq c\log n$ for all large $n$ then $A$ must contain a minimal basis of order $k$? (Here $r(n)$ counts the number of representations of $n$ as the sum of at most $k$ elements from $A$.)
A question of Erdős and Nathanson
[ErNa79], who proved that this is true for $k=2$ if $1_A\ast 1_A(n) > (\log \frac{4}{3})^{-1}\log n$ for all large $n$.
Härtter [Ha56] and Nathanson [Na74] proved that there exist additive bases which do not contain any minimal additive bases.
See also [868].
