Two-intersecting family ($|A\cap B|\ge 2$)

A set family is two-intersecting if $|A\cap B|\ge 2$ for all of its members, including $A=B$. In particular its members all have at least two elements. This is stronger than being an intersecting family.

Fix a member of size $r$. Every other member of a two-intersecting family contains a pair from that fixed member. Counting all $r$-sets containing these pairs gives $|\mathcal{F}|\le \left(\genfrac{}{}{0ex}{}{r}{2}\right)\left(\genfrac{}{}{0ex}{}{n-2}{r-2}\right)$ for a uniform set family.

On an even ground set $n=2h$, a two-intersecting family has size at most ${\sum }_{j=h+1}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)$. On an odd ground set $n=2h+1$, the bound is $\left(\genfrac{}{}{0ex}{}{2h}{h+1}\right)+{\sum }_{j=h+2}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)$. Pair lower levels with complements of upper levels using the intersecting shadow lemma; apply the Erdős-Ko-Rado theorem to the remaining middle level in the odd case.