Problem 870

Erdős Problem #870: declared status 'open'. Formalized: no. 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 $A$.) Prize: no. Tags: additive basis, number theory.