Combinatorial spaces, often related to simple closed curves on surfaces, have been used in different ways to understand Teichmüller spaces, mapping class groups and moduli spaces. More specifically, the curve and pants graphs have been helpful tools to understand geometric properties of Teichmüller spaces with its different metrics and the mapping class group. Flip graphs are other examples of useful combinatorial spaces. The vertices of these graphs are isotopy classes of triangulations and two triangulations share an edge if they are related by a flip (or equivalently differ by a single arc). Flip graphs are also conveniently quasi-isometric to the underlying mapping class groups. The flip graph of a polygon, although finite, has been particularly well studied, most famously by Sleator, Tarjan and Thurston who studied its diameter. In another piece of work, they studied the diameters of flip graphs of punctured spheres (this time up to the action of their mapping class groups) .