Let $ n $ be a positive integer $(n\ge 2)$ and $ A $ be the set $$A=\{1,2,3,...,2n\}$$ Prove that

$$(\forall B\subset A)\;$$ $$ \Bigl(\# B=n+1\implies \exists (a,b)\in B^2 \;:\;a\ne b \wedge a|b\Bigr)$$

It is easy to prove it when $ 1\in B $ and when $ 2\in B $ but i have no idea for the other cases.

Any help will be appreciated. Thanks in advance.

  • $\begingroup$ $b\ne a$ right?? $\endgroup$ – Shubham Johri Nov 8 '20 at 20:42
  • $\begingroup$ Two things, isn’t this trivially true since any number will divide itself and do you mean to say that $|B|=n+1$ (cardinality is $n+1$)? $\endgroup$ – QC_QAOA Nov 8 '20 at 20:43
  • $\begingroup$ @QC_QAOA Yes $ a\ne b $ and $\# $ means Cardinality. Thanks. $\endgroup$ – hamam_Abdallah Nov 8 '20 at 20:47
  • $\begingroup$ For Paul Erdős this was one of his favourite "initiation" questions to mathematics. It is mentioned in the chapter "Pigeon-hole and double counting" in "Proofs from THE BOOK". $\endgroup$ – Hanno Nov 8 '20 at 21:47

Let $B=\{x_1,x_2...x_{n+1}\}\subseteq A$.

Let $x_i=2^{a_i}p_i$ where $2\not|~p_i$ and $a_i\in\Bbb Z_{\ge0}$. Then $p_1,p_2,...,p_{n+1}$ are $n+1$ odd numbers that belong to $S=\{1,3,5,...,2n-1\}$. But $|S|=n$. By the pigeonhole principle, $p_k=p_m$ for some $k\ne m$. This gives $x_k|x_m$ or $x_m|x_k$.

  • $\begingroup$ +1 very nice... $\endgroup$ – user2661923 Nov 8 '20 at 21:27
  • 1
    $\begingroup$ Thank you very much. God bless you. $\endgroup$ – hamam_Abdallah Nov 8 '20 at 21:47
  • $\begingroup$ I searched in vain for alternative (less elegant) proofs. To the best of your knowledge, is there such an alternative proof? $\endgroup$ – user2661923 Nov 9 '20 at 0:06
  • $\begingroup$ @user2661923 I can tell you this is a classic pigeonhole problem. But I am not aware of another proof. $\endgroup$ – Shubham Johri Nov 9 '20 at 15:20

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