Let $S=\{1,2,\cdots,2n\}$, and $S_i \subseteq S(i=1,2,\cdots,n+1)$ be $n+1$ subsets, each of which contains half of the $2n$ elements, namely $|S_i|=n$. Consider the following expression: $$M=\max_{1\le i<j \le n+1} |S_i \cap S_j|$$

• When $n$ is even, it seems that $M\ge n/2$ for any choice of $S_i$. I think it is a really interesting phenomenon, but how does one give a proof of this?
• Also by experiment, I find that$\{(i,j):1\le i<j \le n+1,|S_i \cap S_j|\ge n/2\}$ is $\Omega(n)$, not only one pair.
• This question can be generalized to the following one: Let $S=\{1,2,\cdots,n\}$, and $S_i \subseteq S$ be $m$ subsets of $S$, each of which contains exactly $k$ elements. Can the maximal size of $|S_i \cap S_j|(1\le i<j \le m)$ be bounded from below?
• This question can be generalized further to a real version. Consider $x_i \in \mathbb{R}_+^n$ be $m$ vectors, each of which has a fixed norm: $\parallel {x_i}\parallel _2=k$. Can the maximal inner product of $x_i \cdot x_j(1\le i<j \le m)$ be bounded from below?

Maybe the third and fourth question are hard, but if they can be solved, I think the results will be very useful. Also, the correctness of the first two questions have been checked for a few instances.