Suppose that $(X,<)$ is a dense linear order without endpoints. If $\eta \in 2^{d+1}$, and $\mathcal{C}=\lbrace C \subseteq X : \mathbb{G}(C)=\eta\rbrace$, then $\mathcal{C}$ is $d$-maximum on X.
here $\mathbb{G}(C)$ is genus of $\mathcal{C}$.
To prove this I suppose $X_0 \subseteq X$ to be finite for calculating $\mathcal{C} \vert ^{X_0}$. I try to use induction on $\vert X_0 \vert$ and $d$, but I can't reach any trouble shooter solution. Would be grateful for any help.
Definitions:
suppose $(X,<)$ is a dense and complete linear order, and $B \subseteq X$ is a finite union of convex subsets of $X$. Let $d$ be the number of boundary points of $B$. Define G(B) to be any $\eta \in 2^{d+1}$ such that there are no $a_0 < ··· < a_d$ in $X$ such that $a_i \in B$ if and only if $\eta(i) = 1$, for all $i = 0,...,d$.
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.