From arcs to curves: quadratic growth of 1-systems

We show that the largest size of a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $\chi$ grows quadratically in $\mathrm{\mid }\chi \mathrm{\mid }$. This resolves a longstanding question of Farb-Leininger, up to multiplicative constants. Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of $\mathit{\text{almost}}\mathit{\text{nibs}}$, $\mathit{\text{flowers}}$, and $\mathit{\text{stem}}\mathit{\text{systems}}$ in order to account for how certain polygons built from pairs of curves in the collection distribute their area over the surface.