#17. Medium-Size Intersection Is Hard To Avoid

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Prompts for discussion:

1. Are medium-sized intersections particularly special? Are other-sized intersections also hard to avoid?

2. Here’s the statement that we used in the next miniature, whose proof is in this one, and one that I chose to skip: Let $\mathcal{F}$ be as in the theorem, i.e, $|\mathcal{F}|\le \left(\begin{array}{l}n\\ 0\end{array}\right)+\left(\begin{array}{l}n\\ 1\end{array}\right)+\cdots +\left(\begin{array}{c}n\\ p-1\end{array}\right)$. If $n=4p$, then $\frac{\left(\begin{array}{c}4p\\ 2p-1\end{array}\right)}{|\mathcal{F}|}\ge {1.1}^n\text{. }$ There are (apparently) many ways to prove this. Does this inequality have some visible intuition?