= 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 $\binom{20}2=190$. The complement-pair lower bound is $2\sqrt{\binom{20}5}>210$, establishing strict nonadditivity.
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