Geodesic
Maximal sets of integers with distinct divisors
The electronic journal of combinatorics, Tome 2 (1995)
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A set of positive integers is said to have the 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. Erdős 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 $\beta = \log 2/\log (3/2)$.
DOI : 10.37236/1216
Classification : 11A05
Mots-clés : maximal sets, distinct divisor property 
@article{10_37236_1216,
     author = {R. Balasubramanian and K. Soundararajan},
     title = {Maximal sets of integers with distinct divisors},
     journal = {The electronic journal of combinatorics},
     year = {1995},
     volume = {2},
     doi = {10.37236/1216},
     zbl = {0886.11002},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/1216/}
}
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R. Balasubramanian; K. Soundararajan. Maximal sets of integers with distinct divisors. The electronic journal of combinatorics, Tome 2 (1995). doi: 10.37236/1216

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