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7 - Supersaturated graphs in the edge-triangle model

Published online by Cambridge University Press:  aN Invalid Date NaN

Charles Radin
Affiliation:
University of Texas at Austin
Lorenzo Sadun
Affiliation:
University of Texas at Austin
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Summary

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.

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