Inequalities for quotient densities via grapheur structure

Develop methods using the structure of the space of grapheurs to prove inequalities in quotient densities, or equivalently homomorphism numbers, that hold for graphs of all sizes, in analogy with graphon-based proofs of homomorphism-density inequalities.

Background

The paper proves that any valid continuous inequality in quotient densities for all finite weighted graphs is equivalent to the corresponding inequality over all grapheurs. This transfers the problem from finite graphs to the compact space of grapheur limits.

The unresolved question is whether the structural decomposition of grapheurs can actually be exploited to establish such inequalities, paralleling the role of graphon theory in proving inequalities for homomorphism densities.

References

Can we exploit the structure of the space of grapheurs to prove such inequalities, in analogy with the use of graphon theory to prove homomorphism density inequalities?

— Graph Limits via Quotients  (2512.23149 - Levin et al., 29 Dec 2025) in Section 6, “Conclusions and Future Directions,” item “Inequalities in quotient densities”