Maximal Sets of Integers with Distinct Divisors

Let S denote a set of positive integers and τ : S → be defined so that τ(s) is a proper divisor of s (that is, τ(s) divides s and τ (s) < s). The ensemble (S, τ ) is said to have the ‘distinct divisor property’ if τ is injective, that is, if the τ (s) are different for different values of s. We will also say that S has the distinct divisor property if there exists a τ , as above, such that (S, τ) has the distinct divisor property. Let c > 1 denote a real number and N a large natural number. Let S be a subset of [N, cN ] with the distinct divisor property such that, of all subsets of [N, cN ] having distinct divisors, S has maximal cardinality. If c is fixed and N tends to infinity then P. Erdős and C. Pomerance, [1], have shown that