How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct $A,B,C\in \mathcal{F}$. Must $|\mathcal{F}|=o(2^n)$? Perhaps even$$|\mathcal{F}|<(1+o(1))\left(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor }\right)?$$