Three-color bounds for planar six- and ten-cycles

Prove or disprove that every 3-coloring of the Euclidean plane contains a monochromatic unit-copy of C_6 and a monochromatic unit-copy of C_{10}.

Background

The paper establishes 2-color lower bounds for these cycles but does not determine whether three colors suffice to force a monochromatic copy.

References

Is it true that $2 \xrightarrow{3} C_6$ and $2 \xrightarrow{3} C_{10}$?

— Ramsey problems for graphs in Euclidean spaces and Cartesian powers  (2512.15516 - Axenovich et al., 17 Dec 2025) in Question Q_C6C10, Section 6.1 (Planar case)