Near the boundary of feasible edge and triangle densities, entropy-optimal graphons are unique, multipodal, and analytic in the constraints, yielding infinitely many phases.
Asymptotic Structure of Graphs with the Minimum Number of Triangles
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abstract
We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.
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Emergence in graphs with near-extreme constraints
Near the boundary of feasible edge and triangle densities, entropy-optimal graphons are unique, multipodal, and analytic in the constraints, yielding infinitely many phases.