Consistency of the strong polarized relation at successor and double-successor cardinals

Determine whether the strong polarized relation \(\binom{\kappa^{++}}{\kappa^+}\rightarrow\binom{\kappa^{++}}{\kappa^+}_2\) is consistent with ZFC for some infinite cardinal \(\kappa\).

Background

The paper distinguishes polarized relations on a successor cardinal and its predecessor from the more difficult case in which the two coordinates are a successor and a double successor. It explains that positive relations of the latter form are consistent under AD, but their consistency in ZFC is unresolved.

The results on wondrous ideals provide sufficient conditions for stronger positive polarized relations, while the paper also shows that certain configurations of wondrous ideals over consecutive cardinals cannot exist. These results do not decide whether the displayed two-color relation itself is consistent.

References

An interesting open problem is whether the strong polarized relation is consistent at some pair of successor and double successor cardinals.

— Quadruples and cubes  (2609.11239 - Garti, 10 Sep 2026) in Section 1, Introduction; Section 2, opening discussion of wondrous ideals

A major problem remains open:

\begin{question} \label{qdouble} Is it consistent that \binom{\kappa{++}{\kappa+}\rightarrow\binom{\kappa{++}{\kappa+}_2 holds for some infinite cardinal \kappa \end{question}

— Quadruples and cubes  (2609.11239 - Garti, 10 Sep 2026) in Question 2.12, Section 2, final paragraph