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.