that our (, bipartite) graphons $g$ of Theorem \[near\] maximize the entropy density at least to second order in perturbation theory, among graphons with a limited range of edge and triangle densities. Assuming the $g$ are actually global maximizers we then proved the entropy density would have to lose analyticity as the edge density of the graphon approaches the tripartite regime. We also show that the entropy density must suffer a phase transition as the triangle density is raised sufficiently high, presumably from the structured bipartite phase to a homogeneous phase of higher triangle density.

We expect that a more complicated analysis could produce appropriate graphons $g^{(k)},\ k\ge 1$, near each of the higher edge density (multipartite) graphons of minimial triangle density, with a transition near each scallop intersection. Intuitively this suggests a mechanism whereby as edge density is increased, near the minimum triangle density graphon, the system progressively transitions through finer and finer structure; for high edge density most graphs would consist of many interacting ‘parts’.

Our results on phase transitions require that the graphons of Theorem \[near\] be in fact global, not just local, maximizers of the entropy density. In a future paper [@RS] we use a symmetry 