A strong coloring on a cardinal $\kappa $ is a function $f:[\kappa ]^2\to \kappa $ such that for every $A\subseteq \kappa $ of full size $\kappa $, every color $\unicode{x3b3} <\kappa $ is attained by $f\restriction [A]^2$. The symbol

$$ \begin{align*} \kappa\nrightarrow[\kappa]^2_{\kappa} \end{align*} $$

$\kappa $

We introduce the symbol

$$ \begin{align*} \kappa\nrightarrow_p[\kappa]^2_{\kappa} \end{align*} $$

$f:[\kappa ]^2\to \kappa $

$p:[\kappa ]^2\to \theta $

$A\in [\kappa ]^{\kappa }$

$i<\theta $

$\unicode{x3b3} <\kappa $

$f\restriction ([A]^2\cap p^{-1}(i))$

We prove that whenever $\kappa \nrightarrow [\kappa ]^2_{\kappa }$ holds, also $\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$ holds for an arbitrary finite partition p. Similarly, arbitrary finite p-s can be added to stronger symbols which hold in any model of ZFC. If $\kappa ^{\theta }=\kappa $, then $\kappa \nrightarrow _p[\kappa ]^2_{\kappa }$ and stronger symbols, like $\operatorname {Pr}_1(\kappa ,\kappa ,\kappa ,\chi )_p$ or $\operatorname {Pr}_0(\kappa ,\kappa ,\kappa ,\aleph _0)_p$, also hold for an arbitrary partition p to $\theta $ parts.

The symbols

$$ \begin{gather*} \aleph_1\nrightarrow_p[\aleph_1]^2_{\aleph_1},\;\;\; \aleph_1\nrightarrow_p[\aleph_1\circledast \aleph_1]^2_{\aleph_1},\;\;\; \aleph_0\circledast\aleph_1\nrightarrow_p[1\circledast\aleph_1]^2_{\aleph_1}, \\ \operatorname{Pr}_1(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p,\;\;\;\text{ and } \;\;\; \operatorname{Pr}_0(\aleph_1,\aleph_1,\aleph_1,\aleph_0)_p \end{gather*} $$

$+ \neg $

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 Väänänen from 1994.

We also prove that under $MA(\omega _{1}\!)$, every wide Aronszajn tree weakly embeds in an Aronszajn tree, which combined with a result of Todorčević from 2007, gives that under $MA(\omega _{1}\!)$ every wide Aronszajn tree embeds into a Lipschitz tree or a coherent tree. We also prove that under $MA(\omega _{1}\!)$ there is no wide Aronszajn tree which weakly embeds all Aronszajn trees, improving the result in the first paragraph as well as a result of Todorčević from 2007 who proved that under $MA(\omega _{1}\!)$ there are no universal Aronszajn trees.