Cartesian powers of K3

Characterize the graphs $H$ for which there exists a sufficiently large integer $N$ such that every 2-coloring of $K_3^{\square N}$ contains a monochromatic copy of $H$.

Background

The paper proves this property for layered graphs and notes that it fails for C_4, leaving the general class of graphs unresolved.

References

For which graphs $H$ there exists a sufficiently large integer $N$ such that $K_3{\square N} \xrightarrow{2}H$?

— Ramsey problems for graphs in Euclidean spaces and Cartesian powers  (2512.15516 - Axenovich et al., 17 Dec 2025) in Question, Section 6.5 (Cartesian powers)