USAMO 2007 — Days 1 and 2 Question 3

Let $S$ be a set containing $n^2+n-1$ elements. Suppose that the $n$-element subsets of $S$ are partitioned into two classes. Prove that there are at least $n$ pairwise disjoint sets in the same class.

Proposed by Andras Gyarfas; source discussion: https://aops.com/community/p825499