Here is a question from IMO 2021:
Let $n>100$ be an integer. Ivan writes the numbers $n,n+ 1,\dots,2n$ each on different cards. He then shuffles these $n+ 1$ cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
First find those duplets whose addition will lead to perfect squares.
Next try to form a bipartite graph from those duplets.
Proof to show: A bipartite graph is not possible.
Am I going correctly? Basically, I want to solve this using graph theory. It will be very helpful for me if I get some insight on how to approach this more accurately.