What is the size of the largest subset of integers less than n, such that no number divides any other?

Asked Jul 10 '23 at 04:27

Active Jul 10 '23 at 12:58

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What is the size of the largest subset of integers less or equal than n, such that no number divides any other?

For example if n is 10 we can pick 3,4,5,6,7,10 to get a set of size 6.

Edit: my example is wrong as pointed out in the comments as 3 divides 6 and 5 divides 10

edited Jul 10 '23 at 12:58

Bill Dubuque

asked Jul 10 '23 at 04:27

Teodor Dyakov

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3 divides 6, though. – Misha Lavrov Jul 10 '23 at 04:31
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And $5$ divides $10.$ – aqualubix Jul 10 '23 at 04:34
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Well, it's atleast $\pi(n).$ We can make a subset containing all the primes less than or equal to $n,$ and no element will divide any other element in that subset. – aqualubix Jul 10 '23 at 04:36
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$6,7,8,9,10$ maybe? The largest size is at least $n/2-1$ for even $n$ and $(n+1)/2$ for odd $n$. – Jean-Claude Arbaut Jul 10 '23 at 04:37
For your case of $n=10,$ consider ${2,3,5,7}.$ This is the "prime-subset" I was talking about. – aqualubix Jul 10 '23 at 04:37
@Jean-ClaudeArbaut is right. The elements in the "second-half" of ${1,2,3,\ldots,n}$ cannot divide each other. This idea needs to be made precise though. Also, I am not sure if a bigger subset could somehow exist. Maybe some modular arithmetic would help. – aqualubix Jul 10 '23 at 04:40
And I think it's optimal, using pigeonhole's principle. – Jean-Claude Arbaut Jul 10 '23 at 04:40
@Jean-ClaudeArbaut FYI, you're correct, including that the pigeon-hole principle is one way to prove it, e.g., as shown with the odd part (i.e., with all factors of $2$ removed) of each chosen integer in For a given integer n, what is the cardinality of the greatest subset of (1,n) such that no element of the subset divides another element?, with this being the top item in a site search. – John Omielan Jul 10 '23 at 04:58

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