Products of canonically Ramsey sets

Determine whether products of canonically Ramsey sets are themselves canonically Ramsey.

Background

The paper proves canonical Ramsey results for regular polygons with a prime number of sides and for all powers of such polygons. It then identifies a broader structural question: whether the canonical Ramsey property is preserved under Cartesian products. A positive answer would imply that every canonically Ramsey configuration is contained in arbitrarily large canonically Ramsey configurations and would have consequences for configurations such as {0,1,2}.

References

It is not yet known in general whether products of canonically Ramsey sets are themselves Ramsey---if so, this would imply that all canonically Ramsey sets are contained in arbitrarily large canonically Ramsey sets, and thus are themselves Ramsey.

— The regular pentagon is canonically Ramsey  (2608.19183 - Shaw, 19 Aug 2026) in Section 1, Introduction