Question 11
Let $A$ be a subset of the set $N$ of positive integers. We will say that a subset $B$ of $N$ is a basis of $A$ if the sums of the elements of each non-empty subset of $B$ are different and each element of $A$ is equal to one of these sums. Show that for each $n = 1, 2, 3, \ldots$ there exists a $k(n)$ such that each subset of $N$ with $n$ elements has a base with at most $k(n)$ elements, and determine (for each $n$) the minimum value of $k(n)$.