Martin's axiom and $\omega_1^2 \longrightarrow (\omega_1^2, 3)^2$

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 $ MA_{\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.