Antiferromagnetic homomorphism inequalities conjectures

The conjectures concern lower bounds for homomorphism counts to antiferromagnetic targets, encompassing extremal statements about independent sets, colorings, and semiproper colorings. The central target inequality appears as Conjecture 1.3 in the February 2026 work.

Known results

• The $K_q$ case was previously settled through a generalization of Csikvári’s inequality, as recorded in Zhao’s survey.
• The conjecture was known for $\Delta(G)\leq 2$ and for the target $K_3^\circ$.
• Sah--Sawhney--Stoner--Zhao-type coloring inequalities were proved for several blow-ups of bipartite graphs, including paths, even cycles, and complete multipartite graphs (2025).
• For $q\geq 5$, proper $q$-colorings of $4$-regular graphs were shown to be maximized by unions of $K_{4,4}$ (2018).

August 2026 theorem

Lee, Joonkyung, Seo, and Jaehyeon prove a vertex-inhomogeneous homomorphism inequality for every antiferromagnetic target. The abstract says this unifies the relevant independent-set, coloring, and semiproper-coloring inequalities; it is a corroborated preprint result, not merely an announcement.

Current status (as of August 2026): The unified antiferromagnetic-target inequality is proved in a preprint, apparently resolving the tracked conjectures, although the exact correspondence with every formulation is not specified in the supplied abstract.