We consider a constrained maximization problem
for symmetric measurable functions on [0,1]2
that arises in the study of large random graphs
with constrained edge density and triangle density.
Numerical results in  suggest
that every constrained maximizer g is finite-podal,
meaning that it has only finitely many distinct “vertex types”
g(.,y) as y ranges in [0,1].
Here we prove a weaker property:
excluding strictly negative correlations between edges and two-stars,
the function y→g(.,y)
is constant on some set of positive measure.
As a first step, we show that the vertex types
g(.,y) of g admit a natural order.