We prove the existence of infinitely many distinct phases along the entire boundary of the Razborov triangle. The proofs involve bootstraps, showing that an optimal graphon near the boundary (which always exists by compactness) has to be approximated by a particular graphon using increasingly tight norms, eventually showing that the optimal graphon is unique with a simple structure called multipodal, defined by parameters having a specified asymptotic behavior as we approach the boundary. The "worth" functional, developed in Chapter 2, is a key tool. We then develop order parameters to distinguish these phases. We also relate uniform random graph models to exponential random graph models (ERGMs), showing that due to lack of appropriate convexity almost the entire Razborov triangle is "ERGM-invisible", meaning that one cannot describe graphs with those edge and triangle values using an ERGM. This extends similar results of Chatterjee and Diaconis.

In this short chapter, we establish some basic facts about the edge-triangle model. We review Pikhurko and Razborov’s results about the geography of the phase space of possible values of pairs of the edge and triangle densities, the so-called Razborov triangle. We derive the unique entropy-maximizing graphon at each point on the boundary, and prove that there is a phase transition across the so-called Erdős–Rényi curve in the middle of the phase space. We compute the optimal graphon along a certain line segment that is disjoint from the Erdős–Rényi curve, answering an old question of Chatterjee and Diaconis. Along the way, we develop several proof techniques that will be repeated throughout the book, in particular the Method of Saturated Inequalities and the Method of Competing Inequalities.