Maximum size of intersection of two families of sets

Question

Let $\mathcal A,\mathcal B\in \mathcal P([n])=2^{[n]}$ be two families of subsets of $[n]$. What is the maximal size of $\mathcal A\cap\mathcal B:=\{A\cap B:A\in\mathcal A,B\in \mathcal B\}$ ? I believe the answer is $|\mathcal A|+|\mathcal B|$ but I don't know how to prove it. Could someone help me?

Answer 1

$\newcommand{\A}{\mathcal{A}} \newcommand{\B}{\mathcal{B}} \newcommand{\of}{\subseteq} \newcommand{\P}{\mathcal{P}}$ Consider $n=2m$ (with $m\ge 2$) and define $\A = \{A\of [n] \,:\, A\cap [m] = [m]\}$ and $\B=\{B\of [n]\,:\, B\cap [m+1,n]=[m+1,n]\}$. Then $\A \cap \B = \P([n])$ so $$ |\A \cap \B| = 2^n = |\A| \cdot |\B| \gg |\A| + |\B| = 2^{m+1} $$ The upper bound $|\A\cap \B| \le |\A|\cdot |\B|$ is trivial for all systems $\A$ and $\B$.