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](https://www.renyi.hu/~p_erdos/1961-07.pdf) 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?



Erdős-Ko-Rado with intersections of size at least two
The family sets such that $\lvert A\cap B\rvert\geq t$ for every pair of distinct sets $A,B$ is called *$t$-intersecting*. The harder problem of finding the largest $t$-intersecting $k$-uniform family was solved by [Ahlswede and Khachatrian](http://www.ams.org/mathscinet-getitem?mr=1429238). The problem that you are interested is known as ``Katona's theorem" and admits an easier solution --- see [another paper by Ahlswede and Khachatrian](https://www.math.uni-bielefeld.de/ahlswede/homepage/public/179.pdf).


