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Source. Published pp. 282–283, Proposition 10.1 (PDF).

Statement. If $|A\cap B|=l$ for every $A\in\mathcal A$, $B\in\mathcal B$, then $|\mathcal A||\mathcal B|\le2^n$. Equality holds exactly when $[n]=Y\sqcup Z$, $\mathcal A=2^Y$, and $\mathcal B=2^Z$; then $l=0$.

Proof. Theorem 10.2 gives the bound according to the parity of $l$. Equality is impossible for odd $l$. In the even equality case it also shows that the empty set belongs to both families, forcing $l=0$. Let $Y=\bigcup\mathcal A$ and $Z=\bigcup\mathcal B$. The zero cross intersections imply $Y\cap Z=\varnothing$. Hence

$$|\mathcal A||\mathcal B|\le2^{|Y|+|Z|}\le2^n.$$

Equality forces $Y\cup Z=[n]$ and both families to be the full power sets of their supports. Conversely these two power sets have product $2^n$ and every cross intersection empty. $\square$

Dependencies. theorem_10_2.