Rectangular Partition Relations on $\omega_3$

Definitions and mathematical statement

Let $\omega$ be the first infinite ordinal and $\omega_3$ the initial ordinal of cardinality $\aleph_3$. For a subset $A\subseteq\omega_3$, let $\operatorname{otp}(A)$ denote its order type under the inherited ordinal ordering. A function is constant on a rectangle $A\times B$ if there is an $n\in\omega$ such that $f(a,b)=n$ for every $a\in A$ and $b\in B$.

Is the following assertion consistent with the intended background set theory? \[ \begin{gathered} \exists f:\omega_3\times\omega_3\longrightarrow\omega\quad \forall A,B\subseteq\omega_3,\\ \operatorname{otp}(A)=\operatorname{otp}(B)=\omega \ \Longrightarrow\ \exists a_0,a_1\in A\;\exists b_0,b_1\in B\quad f(a_0,b_0)\ne f(a_1,b_1). \end{gathered} \] The sets $A$ and $B$ need not be disjoint, and no ordering relation between them is imposed. The supplied record does not specify the background axioms or any relative-consistency hypothesis; these remain unresolved qualifications of the consistency question.

Short English statement

Can there consistently be a coloring of $\omega_3\times\omega_3$ by natural numbers with no constant rectangle whose two coordinate sets each have order type $\omega$? The background theory for consistency is unspecified in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001686$, source problem code catalogue entry, titled "Rectangular Partition Relations on $\omega_3$." Collection: Oberwolfach Reports Open Problems; proposed year: 2011.
• Set Theory (2011), Oberwolfach Reports, pp. 85--140, DOI 10.4171/owr/2011/02. Supplied source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2011/02.
• The supplied literature assessment attributes the question to Todorcevic, although the registry's proposer field is empty. It states that the OWR report records the question as open and that a targeted primary-literature search found no later resolution. The exact supplied literature-check date is 2026-08-21. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the problem "open" and the statement corrected_verified, with the supplied explanation that the official report was checked and the statement repaired to match its passage. These are supplied provenance and status claims, not independent verification by this draft.