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Antiferromagnetic models as clique-minimisers

Let GG be a graph, let dv≔deg⁡G(v)d_v\coloneqq\deg_G(v), and let HH be an edge-weighted graph, possibly with loops. Define the weighted homomorphism count by

hom⁡(G,H)≔∑ϕ:V(G)→V(H)∏uv∈E(G)H(ϕ(u),ϕ(v)),\hom(G,H)\coloneqq\sum_{\phi:V(G)\rightarrow V(H)}\prod_{uv\in E(G)}H(\phi(u),\phi(v)),

where H(a,b)H(a,b) is the edge weight between aa and bb. Call HH antiferromagnetic if its symmetric edge-weight matrix has exactly one positive eigenvalue, counting multiplicity. The antiferromagnetic clique-minimiser conjecture. For every graph GG and every antiferromagnetic edge-weighted graph HH,

hom⁡(G,H)≥∏v∈V(G)hom⁡(Kdv+1,H)1dv+1.\hom(G,H)\geq\prod_{v\in V(G)}\hom(K_{d_v+1},H)^{\frac{1}{d_v+1}}.

This conjecture would extend the lower bound for multivariate independence polynomials and its two-spin corollary to all antiferromagnetic homomorphism models. It is established in the paper for the relevant two-spin models, but the general antiferromagnetic case remains open.

References

Primary source

Joonkyung Lee and Jaehyeon Seo, “Lower bounds for multivariate independence polynomials and their generalisations”, arXiv:2602.02450 (2026).

antiferromagnetic-modelsextremal-graph-theorygraph-homomorphisms
Posted

Progress summary

Refreshed
Claimed solved

A complete proof was claimed in August 2026, but it has not been independently verified, so the general conjecture remains unsettled.

The conjecture predicts that, for every graph and every antiferromagnetic weighted target, the weighted homomorphism count is bounded below by the corresponding product of clique counts. Lee and Seo formulated this generalisation in their February 2026 paper, which explicitly presents it as a conjecture.

Known results

  • Lee and Seo (2026) prove the multivariate hard-core-model inequality, covering the multivariate independence-polynomial case.
  • Lee and Seo (2026) prove a stronger two-colour semiproper-colouring inequality, yielding a nontrivial multiaffine special case.
  • Lee, Oh, and Seo (2025) establish related homomorphism inequalities for several antiferromagnetic targets, but not this clique-minimiser conjecture.
  • Earlier work proves clique extremality for the Widom–Rowlinson model and related special models.

Posted attempt

In August 2026, Joonkyung Lee and a coauthor claimed to have uploaded a complete proof. The attempt has not been independently verified, and no corroborating source was retrieved.

Current status (as of August 2026): Special cases and related inequalities are established, while a complete proof of the general antiferromagnetic clique-minimiser conjecture is only claimed and remains unverified.

Sources

Solutions 1

ProofThis solution needs a summarySee full solutionHide full solution

Joonkyung Lee and I just uploaded a proof to arXiv: see https://arxiv.org/abs/2608.17920 .

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Jaehyeon Seo Posted