In this short chapter, we show that the edge-kstar model exhibits many of the properties of the edge-2star model. In particular: all entropy maximizers are multipodal with a uniform bound on the number of defining parameters; there is a single bipodal phase along the lower boundary of the phase space; near the top of the phase space there is a phase transition between cliquelike and anticliquelike graphs, with two optimal graphons on the curve dividing these regions. We also show that there is a bipodal phase in a model where we constrain an arbitrary positive linear combination of kstar densities. As we will see in Chapter 7, every edge-H model (where H is a fixed graph) is well approximated by an edge-(linear combination of kstars) model. The estimates of this chapter are at the core of the proof of Theorem 7.11, proving the existence of a bipodal phase for all edge-H models.

The main result of this chapter is a proof of the existence of a bipodal phase just above the entire length of the curve bisecting the Razborov triangle. This requires separate arguments for e < ½, e > ½, and e approximately ½, with the arguments governing overlapping regions whose union stretches all the way from e = 0 to e = 1. We previously proved the existence of a bipodal phase just below the top boundary of the Razborov triangle in Chapter 4. It is likely, but as yet unproven, that these two regions are part of a single F(1,1) phase covering the entire region where $t>e^3$. We then show that these results are not specific to the edge-triangle model. In every "edge-H" model where we constrain the densities of edges and one other graph H, we prove that there is an F(1,1)-like bipodal phase lying above all but finitely points of the Erdős–Rényi curve.