A problem with 26 distinct positive integers

Apply Dilworth's Theorem to the poset of the $26$ integers under the divisibility relation. Either there is an antichain of length $6$ (no two of $6$ integers divide one another) or the set can be partitioned into at most $5$ chains (sequences where each integer divides the next), one of which must have length at least $6$ by the pigeonhole principle.

Define a chain as a sequence $(y_j)$ such that $y_1|y_2 |...|y_k$ and an antichain a sequence $(y_j)$ such that $y_1 \nmid y_2 \nmid .. \nmid y_k$.

To each $x$ in the initial sequence attach a pair of numbers $(i_x,j_x)$ such that $i_x$ is the the length of the longest chain ending in $x$ and $j_x$ is the length of the longest antichain starting with $x$.

Pick now two elements $x\neq y$ such that $x$ is before $y$ in the sequence $(x_n)$. If $x | y$ then $i_x<i_y$ since any chain ending in $x$ can be extended to a longer chain ending in $y$. If $x \nmid y$ then $j_x > j_y$ since any antichain starting in $y$ can be extended to a longer antichain starting in $x$. Therefore the application $x \mapsto (i_x,j_x)$ is injective.

Consider a sequence with $mn+1$ elements, and suppose the length of the longest chain is $m$ and the length of the longest antichain is $n$. Therefore the map $x \mapsto (i_x,j_x)$ is an injection from a set with $mn+1$ elements into a set with $mn$ elements. This is a contradiction proving that there is a chain of length $m+1$ or an antichain of length $n+1$.

In the problem presented here we have $m=n=5$.

I have found this solution (for 17 numbers though) in a Russian site, as this problem was a question in a 1983 Soviet Mathematics contest (Турниры Городов) for student of 7-8 classes!

Solution. We shall attach on each of our numbers $$ x_1<x_2<\cdots<x_{26}, $$ an index next to their subscript in the following fashion:

$x_1$ becomes $x_{1,1}$.

If $x_1$ divides $x_2$, then $x_2$ becomes $x_{2,2}$, otherwise it becomes $x_{2,1}$.

In general, if we have attached indices to $x_1,\ldots,x_k$, then the index of $x_{k+1}$ will be $1$ is none of the $x_1,\ldots,x_k$ divides $x_{k+1}$, or it will become $i+1$, if $i$ is the largest index of all the numbers which divide $x_{k+1}$.

If the index of some of the $x_j$'s is at least $6$ then we have have a sequence $$ x_{k_1,1} \mid x_{k_2,2} \mid x_{k_3,3} \mid x_{k_4,4} \mid x_{k_5,5}\mid x_{k_6,6}. $$ In all the indices are less or equal to $5$, then some number in $\{1,2,3,4,5\}$, say $j$, is necessarily the index of at least $6$ numbers, i.e. $$ x_{k_1,j} < x_{k_2,j} < x_{k_3,j} < x_{k_4,j} < x_{k_5,j} < x_{k_6,j}, $$ which means that none of the above $6$ numbers divide another one of them.

Note. Τhis can be generalised to finding a chain of $k$ numbers, fully ordered by division, or a subset of $\ell$ numbers, with none of them dividing another one, among $m$ distinct positive integers, whenever $(k-1)(\ell-1)\le m-1$.