Unresolved Ramsey-equivalence question for connected graphs

Determine whether there exist non-isomorphic connected graphs H_1 and H_2 that are Ramsey-equivalent, meaning that \mathcal{M}_2(H_1)=\mathcal{M}_2(H_2).

Background

The paper recalls an unresolved question from the Ramsey-equivalence literature concerning non-isomorphic connected graphs with identical families of minimal 2-Ramsey graphs. This question is distinct from the paper’s subsequently solved Maker–Breaker analogue.

References

The problem is still unsolved.

— Ramsey-type results for Maker-Breaker games  (2609.02588 - Allin et al., 2 Sep 2026) in Subsection “Equivalent graphs for cliques,” Concluding remarks and open problems

A main motivation of this note is a question by Fox, Grinshpun, Liebenau, Person, and Szabo , whether any two connected non-isomorphic graphs have distinct edge Ramsey classes. While this question remains open in general, we answer the analogous question in the positive for vertex colourings and all, not only connected, graphs.

— Non-isomorphic graphs have distinct vertex-Ramsey classes  (2610.03704 - Axenovich et al., 2 Oct 2026) in Section 1, Introduction