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Here's a little question that we were shown in class:

Let $S = \{1,2,\ldots,200\}$ and let $A \subseteq S$ such that $|A| = 101$. Prove that there are two elements of $A$ such that one is a divisor of the other.

The proof was fairly easy: the pigeonholes were the pattern $(2k+1)2^i$. You have a $100$ of those and therefor two elements must be in the same pigeonhole.

My question: Why a $100$ pigeonholes? Other than that face that this is comfortable and it works, is there any other reason? Can I do the same for $50$ and still be able to give such representation to every number?


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In (2k+1)a^i, you mean a=2, right? – Lopsy Dec 26 '11 at 17:34
Yes, indeed. I've fixed it – yotamoo Dec 26 '11 at 18:04
For this problem 100 is the "correct" answer: there exists some $A \subseteq S$ of size 100 such that no element of it divides another. (Exercise: Find such an $A$.) I do not know what you mean by "representation", but certainly you would not be able to prove that 50 is an upper bound. // This problem is also discussed elsewhere in this site: – Srivatsan Dec 26 '11 at 18:25
Concerning the sentence "My question: Why a $100$ pigeonholes? Other than that face that this is comfortable and it works, is there any other reason?": Srivatsan's comment aside; there is no general algorithm that solves any problem you come up with in a sleeping carriage. – Christian Blatter Dec 26 '11 at 19:54
up vote 2 down vote accepted

Well, you're absolutely right; if you could partition $S$ up into $50$ subsets such that every member of every subset was either a divisor or a multiple of every other element in the same subset, then you'd have proved the even stronger statement that any $A \subseteq S$ with $|A| = 51$ would contain divisor-dividend pair.

But you can't make that partition. To show this, we again use the pigeonhole principle: suppose that you have fewer than 100 pigeonholes. Then some two of the numbers from $101$ to $200$ will be in the same pigeonhole. This pigeonhole thus fails to contain only numbers that divide each other.

So a $101$-element subset is in fact not only sufficient but also necessary to guarantee that it has two elements with one dividing the other.

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