Let $S$ be a compact orientable 2-surface with $\chi(X)\leq -3$. Y.Minsky and H.Masur proved that the curve complex of $X$ is $\delta$-hyperbolic and infinite.
E.Klarreich and Ursula Hamenstadt proved the gromov boundary of the curve complex of $S$ is bijective to the 
collection of minimal filled lamination. Denote the collection of minimal 
filled lamination $B$.

Note: filled lamination means its complement in $S$ is either a polygon or one-punctured polygon. 

My question is :
Given a number $N$, is there possible that there is a collection of essential simpple closed curves $Y=\{c_{\eta}, \eta\in B\}$ such that $d_{\mathcal {C}(S)}(c_{\eta}, c_{\zeta})\geq N$, for any $\eta, \zeta \in B$ and $\eta\neq \zeta$?