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Level C · Reviewed Logic & formalisation P-graph-density-binomials-decidability

Is t(G1) ≥ t(G2) decidable for graph homomorphism densities?

Linear inequalities between graph homomorphism densities are undecidable in general. Is the simplest case — deciding whether t(G1) ≥ t(G2) holds for all graphons — decidable?

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Cite 
@misc{cairn-graph-density-binomials-decidability,
  title        = {Is t(G1) ≥ t(G2) decidable for graph homomorphism densities?},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/graph-density-binomials-decidability}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
}

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The problem

The question

For finite graphs G and W, t(G, W) is the homomorphism density of G in W (extended to graphons). A binomial of graph densities is a₁·t(G₁) + a₂·t(G₂) with integer coefficients; the interesting case is deciding whether t(G₁, W) ≥ t(G₂, W) holds for every graphon W. Is this decidable?

What is known

Hatami and Norine showed that deciding general linear inequalities between homomorphism densities is undecidable. Related undecidability results exist for tournaments and other settings. The binomial case is open.

What counts as progress

  • Decidability for subclasses (e.g. G₁, G₂ forests, or bounded size), or a reduction showing undecidability.
  • Certified individual instances (sum-of-squares or flag-algebra certificates; counterexample graphons).

Source. Posed by Greg Blekherman in the open problem session of the Oberwolfach workshop Proof Complexity and Beyond (2024), recorded in Oberwolfach Reports 15/2024, p. 875 (EMS Press, DOI 10.4171/OWR/2024/15), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.