I am having a hard time thinking about how to cast the below problem into a form that can clearly make use of the Pigeonhole Principle.
Suppose we are given a set $T = \lbrace 1, \cdots, k \rbrace$ for a positive even integer $k$. I would like to show that for any $S \subseteq T$ such that $|S| = \frac{k}{2} + 1$, there exists $x,y \in S$ with $x \neq y$ such that $x$ divides $y$.
The most basic pigeonhole principle basically assumes we have $n+1$ "elements" that we would like to insert into $n$ "boxes" and the principle tells us that we can expect at least 2 elements to be in at least one of the boxes. This principle can be used to prove a lot of things but most examples I have found have pretty obvious "elements" and "boxes", while in this case I am not seeing anything obvious.
It seems at first that one would want to perform $y \mod x$ for each valid pair of $x,y \in S$, but this does not seem smart because then it is not clear what the buckets should be since using $k$ buckets from $0$ to $k-1$ seems not really great when the modulo is done for values that may not be $k$.
Can anyone provide a hint as to how I might approach the problem using this technique?
