We show the existence of two bipodal phases with $t<e^3$, the B(1,1) phase near the Erdős–Rény curve and the A(2,0) phase near e = ½. Combined with our previous proof of the F(1,1) phase, the B(1,1) result gives a complete description of both sides of the Erdős–Rényi phase transition for e > ½. We then show that the optimal graphon for e close to ½ and $t < e^3$ is symmetric bipodal and construct an order parameter to distinguish this phase from B(1,1). However, the A(2,0) phase does not cover the entire region where e < ½ and $t<e^3$. We prove that when e < 0.2113 and t is slightly less than $e^3$, the entropy maximizing graphon is not bipodal. Generalizing the triangle constraint, we consider edge-H models, especially where H is an odd cycle, and also general H, and prove several properties.