  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.
