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Suppose $\{a_n\}_{n=1}^{m^2+1}$ is a strictly increasing sequence of $n^2+1$ positive integers, show that there exist a subsequence $\{a_{i_b}\}_b^{n+1}$ of length $n+1$ such that $a_{i_k}$ is divisible by $a_{i_j}$ for any $1\le j<k\le n+1$ or $a_{i_k}$ is not divisible by $a_{i_j}$ for any $1\le j<k\le n+1$.

I have tried some small $n$ like when $n=2$ and tried a few examples of sequences of positive integers and all consisted of such sub-sequence but i have no idea how to prove the general case, that is for all $n$

Local L
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1 Answers1

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It is answered here in the case $n=5$. The general case is dealt with in the same way.

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Yiorgos S. Smyrlis
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