On wide Aronszajn trees in the presence of MA

Abstract: A wide Aronszajn tree is a tree of size and height $\omega_1$ with no uncountable branches. We prove that under $MA(\omega_1)$ there is no wide Aronszajn tree which is universal under weak embeddings. This solves an open question of Mekler and V\"a\"an\"anen from 1994. We also prove that under the same assumption there is no universal Aronszajn tree, improving a result of Todor\v{c}evi{\'c} from 2007 who proved the same under the assumption of BPFA for posets of size $\aleph_1$. Finally, we prove that under $MA(\omega_1)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree.