Existence of a symmetric bipodal phase in the edge-triangle model

In the edge-triangle model with edge density close to 1/2 and triangle density below 1/8 we prove that the unique entropy-maximizing graphon is symmetric bipodal. We also prove that, for any edge density e less than e0=(3-3)/6 approximate to 0.2113 and triangle density slightly less than e 3, the entropy-maximizing graphon is not symmetric bipodal. We also discuss the implications for an old idea of Landau for using symmetry to give an intrinsic difference between solid and fluid phases of matter.