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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?

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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. The problem that you are interested is known as ``Katona's theorem" and admits an easier solution --- see another paper by Ahlswede and Khachatrian.

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