Nonadditivity of Shannon capacity

Capacity of a disjoint union can strictly exceed the sum of the capacities. On the five-element subsets of a twenty-point set, join distinct sets when their intersection is odd. Linear polynomials over ${\mathbb{F}}_2$ bound its capacity by twenty; squarefree quadratic polynomials over ${\mathbb{F}}_3$ bound its complement's capacity by $\left(\genfrac{}{}{0ex}{}{20}{2}\right)=190$. The complement-pair lower bound is $2\sqrt{\left(\genfrac{}{}{0ex}{}{20}{5}\right)}>210$, establishing strict nonadditivity.