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Abstract for R. Balasubramanian  and  K. Soundararajan, Maximal Sets of Integers with Distinct Divisors

 A set of positive integers is said to have the {\sl
distinct divisor property} if there is an injective map that
sends
every integer in the set to one of its proper divisors.  
In 1983, P.~Erd{\H o}s and C.~Pomerance showed that for
every $c>1$, a largest subset of $[N,cN]$ with the
distinct divisor property has cardinality $\sim \delta(c)N$,
for some constant  $\delta(c)>0$. They conjectured  that  
$\delta(c)\sim c/2$ as $c \to \infty$.
We prove their conjecture. In fact we show that there exist  
positive absolute constants $D_1,D_2$ such that
$D_1\le c^{\beta}(c/2-\delta(c))\le D_2$ where
$\beta = \log 2/\log (3/2)$.


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