Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 109 / 4 / ii / Solution

Fix $A\in \mathcal{F}$ and consider the tracesFor distinct $B,C\in \mathcal{F}\setminus \{A\}$, the hypothesis applied in both orders saysEquivalently, neither trace contains the other. Thus the traces are distinct and form an antichain in the Boolean lattice on the $2r$ points of $A$. By Sperner theorem,which is the required bound.