I am trying to solve the following problem.
Assume that we are given 26 distinct positive integers. Show that either there exist 6 of them $x_1<x_2<x_3<x_4<x_5<x_6$, with $x_1$ dividing $x_2$, $x_2$ dividing $x_3$, $x_3$ dividing $x_4$, $x_4$ dividing $x_5$ and $x_5$ dividing $x_6$ or there exist six of them, such that none of them divides another one of these six.
A possibly good start is to assume that, in every six of these numbers, there exists at least one dividing another one of the same six.
Update. I have found a solution of the problem (for 17 numbers though) in a Russian site. As unbelievable as it may sound, this problem was a question in a 1983 Soviet Mathematics contest (Турниры Городов) for student of 7-8 grades!
I am presenting the solution I found in that site below as an answer, and it is generalised for $n^2+1$ distinct integers, where we show that either there exist $n+1$ of them dividing each other or none diving none else.