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

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$)+2^{\aleph_0}=\aleph_2 +\omega_1^2\nrightarrow(\omega_1^2,3)^2.$Wealsoisolateasimpleforcing-preservationprinciple:thesameground-modelcoloringremainsawitnessafterforcingwithanyposetwhosesubfamiliesofsizeatmost$\omega_1$arecountableunionsoflinkedsets\mathrm{.}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.