Filling systems of maximum size

Let $S_g$ be a closed orientable surface of genus $g\geq 2$. A collection $\Omega = \{ \gamma_1, \dots, \gamma_s\}$ of pairwise non-homotopic simple closed curves on $S_g$ such that $\gamma_i$ and $\gamma_j$ are in minimal position, is called a \emph{filling system} or a \emph{filling} of $S_g$ if the complement $S_g\setminus \Omega$ is a disjoint union of $b$ topological discs for some $b\geq 1$. The \emph{size} of a filling system is defined as the number of its elements. We prove that the maximum size of a filling system on $S_g$ with $1 \leq b \leq 2g-2$ boundary components is $2g+b-1$. Furthermore, we give a lower bound on mapping class group orbits of filling systems of maximum size with $1 \leq b \leq g-2$ boundary components.