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Let $S=\{1,2,3,\ldots, 2n\}.$ If $P\subseteq S$ such that if $x,y\in P \rightarrow x\nmid y$ and $y \nmid x$ Then what is the maximum cardinality of $P$

I know that $\{n+1, n+2,\ldots, 2n\}\subseteq P$ so $|P|\geq n$

How can I proceed from here. I somehow want to use Pigeonhole principle.

So Lo
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  • $n\mid 2n$, so your $P$ doesn't work. However, if you remove $n$ from your $P$, you still have $n$ elements. – Arthur Nov 18 '19 at 13:07
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    This is sort of a duplicate. I'm still looking for a better match, though with what you have (after Arthur's correction) that answers your question. – Daniel Fischer Nov 18 '19 at 13:18
  • A different approach; sort of pigeon hole, with guidance. – Daniel Fischer Nov 18 '19 at 13:28
  • "I know that ${n,n+1,\dots,2n}\subseteq P$" Not necessarily. Consider $n=2$ so $S={1,2,3,4}$. There are two examples of subsets of maximum cardinality each of which satisfying the desired property, namely ${2,3}$ and ${3,4}$. It is a particularly strong claim, one that requires proof, to say that ${n+1,n+2,\dots,2n}$ has the maximum cardinality of such a set. Certainly, you could show that it is maximal in the sense that you couldn't add any more to it, but what makes it so that a set like ${2,3,5,7,\dots}$ wouldn't have more elements than ${n+1,n+2,\dots,2n}$? – JMoravitz Nov 18 '19 at 13:29
  • sorry I meant ${n+1,n+2\ldots 2n}$ – So Lo Nov 18 '19 at 15:58

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