It is consistent with MA_aleph_1(sigma-centered) plus continuum aleph_2 that omega_1^2 does not arrow (omega_1^2,3)^2, and under full MA_aleph_1 every c.c.c. forcing preserves such witnesses.
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Martin's axiom and $\omega_1^2 \longrightarrow (\omega_1^2, 3)^2$
It is consistent with MA_aleph_1(sigma-centered) plus continuum aleph_2 that omega_1^2 does not arrow (omega_1^2,3)^2, and under full MA_aleph_1 every c.c.c. forcing preserves such witnesses.