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Let $A \subset \mathbb N$ be a antichain with respect to divisibility. Does this imply that the density of $A$ is $0$?

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Since it is related to (and likely no greater than, although this most likely your point) the density of the set of primes, and (for some notions of density) the primes have density 0, I'd guess no, such antichains do not have positive density. Gerhard "Ask Me About System Design" Paseman, 2011.08.29 –  Gerhard Paseman Aug 29 '11 at 15:38
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up vote 8 down vote accepted

Such a subset is called "primitive"; such a set must have lower density zero but (as shown by Besicovitch) does not need to have asymptotic density zero. Check out the references and results of this recent paper of Greg Martin and Carl Pomerance:

http://arxiv.org/pdf/1009.1014v2

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