Erdős problem 56

Suppose $A \subseteq \{1,\dots,N\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime. An example is the set of all multiples of the first $k$ primes. Is this the largest such set? To avoid trivial counterexamples, we must insist that $N$ be at least the $k$th prime.