to prove that these graphons are indeed the unique global maximizers at least for triangle density in the range $0\le t\le 1/8$ and edge density $e=1/2$, and we can then see a transition on this curve. However we still cannot prove the graphons are the global optimizers of entropy density for $(e,t)$ in any two-dimensional region, as is needed to fully justify the notion of a structured phase. (See [@AR] for a variant of this approach.)

In conclusion we emphasize that our key tool was Theorem \[thm1\], an optimization formula for the asymptotic entropy density, and made essential use of the graph limit formalism. The graphon formalism is a powerful tool for dealing with the infinite size limit in mean field models, and we have used it to make some progress on understanding the structure of asymptotically large graphs near the extreme of low triangle density.

[**Acknowledgements:**]{} We gratefully acknowledge useful discussions with Francesco Maggi and Peter Winkler.

[BCLSV]{}

D. Aristoff and C. Radin, Emergent structures in large networks, J. Appl. Probab. (to appear), arXiv:1110.1912

, Moments of two-variable functions and the uniqueness of graph limits, Geom. Funct. Anal. 19 (2010) 1597-1619.

, Convergent graph sequences I: subgraph frequencies, metric 