Martin's axiom and $ω_1^2 \longrightarrow (ω_1^2, 3)^2$

Abstract: Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $σ$-centered forcing notions gives a model of $MA_{ω<em>1}(σ$-centered$$)+2^{{\aleph_0}=\aleph_2} +ω_1<sup>2\nrightarrow(ω_1<sup>2,3)<sup>2.$$ We also isolate a simple forcing-preservation principle: the same ground-model coloring remains a witness after forcing with any poset whose subfamilies of size at most $ω_1$ are countable unions of linked sets. Under $\text{MA}</em>{ω_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcing over that model.

• The paper proves that locally ℵ₁-σ-linked forcing preserves any coloring witnessing the failure of ω₁² → (ω₁², 3)², using the σ-ideal of short subsets of ω₁².
• Under MA_{ℵ₁}, every c.c.c. forcing is locally ℵ₁-σ-centered, so no c.c.c. extension can destroy an existing counterexample to the partition relation.
• A finite-support iteration establishes consistency of MA_{ℵ₁}(σ-centered) + 2^{ℵ₀} = ℵ₂ with the negative relation and 𝔟 = 𝔰 = ℵ₂, while leaving the full MA_{ℵ₁} question open.

Background and problem

The paper studies the ordinal partition relation $\omega_1^2 \longrightarrow (\omega_1^2, 3)^2$, a question in the Erdős–Hajnal–Rado partition calculus. The relation asserts that every graph on $\omega_1^2$ (the ordinal product, not merely the cardinal $\aleph_1$) contains either an independent set of order type $\omega_1^2$ or a triangle (clique of size 3). Hajnal showed in 1971 that CH implies the failure of this relation, via a set-mapping construction organized by an enumeration of small configurations at length $\aleph_1$ [Hajnal1971]. Larson later weakened the hypothesis to the existence of a short scale, showing $\mathfrak{b} = \aleph_1$ implies the failure [Larson1998].

The open problem motivating the paper is due to Erdős and Hajnal: does $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$ imply $\omega_1^2 \to (\omega_1^2, 3)^2$? Baumgartner posed the analogous question for PFA. Baumgartner had earlier obtained positive forcing-axiom results for smaller ordinals, e.g. $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$ under a suitable form of Martin's Axiom [Baumgartner1989]. The full $MA_{\aleph_1}$ question remains open; this paper contributes a preservation theorem and a partial consistency result.

The $\omega_1^2$0-ideal of short subsets

Let $\omega_1^2$1, with canonical rows $\omega_1^2$2. The key structural lemma characterizes membership in $\omega_1^2$3: a set $\omega_1^2$4 has order type below $\omega_1^2$5 if and only if the set of rows meeting $\omega_1^2$6 in $\omega_1^2$7 many points is countable. Consequently $\omega_1^2$8 is a $\omega_1^2$9-ideal: a countable union of short sets is short, because the "large row" sets of the components are countable and their union remains countable.

The author emphasizes that closure under countable unions—not merely finite unions—is essential to the forcing argument. This property is specific to the two-level ordinal structure $\aleph_1$0.

Preservation under locally $\aleph_1$1-$\aleph_1$2-linked forcing

The central technical notion is local $\aleph_1$3-$\aleph_1$4-linkedness: every subfamily of size at most $\aleph_1$5 is a countable union of linked sets. Every $\aleph_1$6-linked forcing has this property trivially.

Preservation theorem. If $\aleph_1$7 witnesses the failure of $\aleph_1$8 in the ground model, then any locally $\aleph_1$9-$\omega_1^2$0-linked forcing preserves $\omega_1^2$1 as a witness in every generic extension.

The proof is a direct reflection argument. If some condition forced a name for a color-0 homogeneous set $\omega_1^2$2 of order type $\omega_1^2$3, one collects conditions $\omega_1^2$4 forcing each $\omega_1^2$5 into $\omega_1^2$6. The resulting index set $\omega_1^2$7 cannot be short, since order types do not increase under forcing. Local $\omega_1^2$8-$\omega_1^2$9-linkedness decomposes $\aleph_1$0 into countably many linked families; since $\aleph_1$1 is a $\aleph_1$2-ideal, one family indexes a non-short $\aleph_1$3, and compatibility forces $\aleph_1$4 to be a ground-model 0-homogeneous set of order type $\aleph_1$5—contradicting the choice of $\aleph_1$6.

Under $\aleph_1$7, every c.c.c. forcing is locally $\aleph_1$8-$\aleph_1$9-centered: given $\mathfrak{b} = \aleph_1$0 of size at most $\mathfrak{b} = \aleph_1$1, the Boolean subalgebra of $\mathfrak{b} = \aleph_1$2 generated by $\mathfrak{b} = \aleph_1$3 has size at most $\mathfrak{b} = \aleph_1$4, is c.c.c., hence $\mathfrak{b} = \aleph_1$5-centered by the standard consequence of Martin's Axiom that c.c.c. forcings of size $\mathfrak{b} = \aleph_1$6 are $\mathfrak{b} = \aleph_1$7-centered under $\mathfrak{b} = \aleph_1$8. Pulling the centered pieces back yields the required decomposition. Combining these facts:

Corollary. Under $\mathfrak{b} = \aleph_1$9, every c.c.c. forcing preserves every existing witness to the failure of $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$0.

This is a strong rigidity statement: once such a witness exists in a model of $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$1, no c.c.c. extension can destroy it. Any proof that $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$2 implies the positive relation must therefore show that no ground model satisfying those hypotheses can carry a witness at all—it cannot proceed by a c.c.c. destruction argument over a witnessing model.

Consistency of the negative relation with restricted MA

The second main result gives a partial answer to Erdős and Hajnal's question:

Theorem. Relative to ZFC, it is consistent that

$MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$3

and moreover $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$4.

Starting from GCH, fix Hajnal's coloring $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$5 and perform a finite-support iteration of length $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$6 of $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$7-centered forcings, with Cohen forcing at cofinally many stages, bookkeeping all names for $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$8-centered forcings with $MA_{\aleph_1} + 2^{\aleph_0} = \aleph_2$9 many dense sets. The standard iteration fact—that a finite-support iteration of $\omega_1^2 \to (\omega_1^2, 3)^2$0-centered forcings of length strictly below $\omega_1^2 \to (\omega_1^2, 3)^2$1 is $\omega_1^2 \to (\omega_1^2, 3)^2$2-centered when $\omega_1^2 \to (\omega_1^2, 3)^2$3 is computed in the ground model—implies each initial segment $\omega_1^2 \to (\omega_1^2, 3)^2$4 ($\omega_1^2 \to (\omega_1^2, 3)^2$5) is $\omega_1^2 \to (\omega_1^2, 3)^2$6-centered, since $\omega_1^2 \to (\omega_1^2, 3)^2$7 in the ground model. Any subfamily of the final poset of size $\omega_1^2 \to (\omega_1^2, 3)^2$8 appears inside some $\omega_1^2 \to (\omega_1^2, 3)^2$9, so the final forcing is locally $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$0-$$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$1-centered, and the preservation theorem ensures $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$2 survives. The cardinal equation $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$3 follows from $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$4.

Combining this with Larson's theorem yields a methodological corollary: it is consistent that $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$5 while the negative relation holds, so the converse implication "$$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$6 implies $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$7" is not provable in ZFC. In other words, the short-scale hypothesis in Larson's result is sufficient but not necessary.

Limitations and open questions

The main Erdős–Hajnal question—whether full $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$8 implies $$\omega \cdot \omega_1 \to (\omega \cdot \omega_1, 3)^2$$9—remains open, as does Baumgartner's PFA variant. The consistency result only covers the $MA_{\aleph_1}$0-centered fragment of Martin's Axiom; the preservation machinery applies to locally $MA_{\aleph_1}$1-$MA_{\aleph_1}$2-linked posets, and while $MA_{\aleph_1}$3 upgrades all c.c.c. posets to this class, the paper does not establish whether witnesses exist in models of the full axiom, nor whether proper or non-c.c.c. forcings can destroy a witness. The argument also relies on the specific $MA_{\aleph_1}$4-ideal structure of $MA_{\aleph_1}$5; extending the technique to other ordinals $MA_{\aleph_1}$6 would require analogous ideal properties that are not established here.

Conclusion

The paper isolates local $MA_{\aleph_1}$7-$MA_{\aleph_1}$8-linkedness as a forcing-preservation principle for witnesses to $MA_{\aleph_1}$9, proves that under $\omega_1^2$00 all c.c.c. forcings have this property, and derives the relative consistency of the negative relation with $\omega_1^2$01, $\omega_1^2$02, and $\omega_1^2$03. These results constrain any future affirmative solution to the Erdős–Hajnal problem and show that Larson's scale-based criterion cannot be reversed in ZFC.