Up to how many subsets of $\{1,2,\dots,2n\}$ of size $n$ can we choose so that each pair has an intersection of size at least two?
The original Erdős-Ko-Rado paper shows that taking all subsets that contain two fixed elements (thus yielding $\binom{2n-2}{n-2}$ subsets) is not necessarily optimal, but doesn't show what is optimal. What is known about this problem?