Quantum theory realizes all joint measurability graphs

Joint measurability of sharp quantum observables is determined pairwise, and so can be captured in a graph. We prove the converse: any graph, whose vertices represent sharp observables, and whose edges represent joint measurability, is realized by quantum theory. This leads us to show that it is not always possible to use Neumark dilation to turn unsharp observables into sharp ones with the same joint measurability relations.