How we can prove that if $C_n$ is the class containing all the unions of at most $n$ positive intervals on the line. Then $C_n$ is $2n$ maximum.
Is it also true when we consider all the unions of exactly n intervals on the line?
Say that $\mathcal{C} \subseteq 2^X$ is $d$-maximum for $d \in \omega$ if for any finite $A \subseteq X$, $|\mathcal{C}|^A| := |\big\{C \cap A : C \in \mathcal{C}\big\}| = \sum_{i=0}^d {|A|\choose{i}}$ if $d < |A|$ and $2^{|A|}$ otherwise.