Martin's axiom and ${\omega }_1^2⟶({\omega }_1^2,3{)}^2$

Abstract

Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $\sigma$-centered forcing notions gives a model of $M{A}_{{\omega }_1}(\sigma$-centered$)+2^{{\aleph }_0}={\aleph }_2+{\omega }_1^2\nrightarrow ({\omega }_1^2,3{)}^2.$ 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 ${\omega }_1$ are countable unions of linked sets. Under ${\text{MA}}_{{\omega }_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcing over that model.