Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 109 / 2 / ii / Solution

Use the four families from the hint. Put $N=2^n$, $u=|{\mathcal{U}}_1|/N$, and $v=|{\mathcal{V}}_1|/N$. The families ${\mathcal{U}}_1,{\mathcal{V}}_1$ are up-sets, while their complements ${\mathcal{U}}_2,{\mathcal{V}}_2$ are down-sets. Applying the Harris-Kleitman inequality to the up-sets, and equivalently to the down-sets after taking complements of the ground set, givesThe cross-incomparability assumption givesConsequentlyThe Cauchy-Schwarz inequality now yieldsThis is the sharp two-family Cross-Sperner inequality.