TITLE: larger coherent family of functions
QUESTION [7 upvotes]: One way to construct an Aronszajn tree is to build a sequence of functions $\langle e_\alpha : \alpha < \omega_1 \rangle$ such that $e_\alpha$ is an injection from $\alpha$ to $\omega$, and for any $(\alpha, \beta)$, $e_\alpha$ agrees with $e_\beta$ except on a finite set.
Is the following generalization possible?  There is a family of functions {$e_x : x \in [\omega_2]^\omega$} such that each $e_x$ is an injection from $x$ to $\omega$, and for any $x,y \in [\omega_2]^\omega$, {$\alpha : e_x(\alpha) \not= e_y(\alpha)$} is finite.

REPLY [6 votes]: To add to Joel's answer; you can construct such a family, so long as you're willing to relax the injectivity requirement to just "finite-to-one".  See P. Koszmider, "On Coherent Families of Finite-to-One Functions", JSL 58 (1993) no. 1, 128-138.  It's in JSTOR.
Koszmider proves that you can construct such a family (with functions that are finite-to-one) on any $[\omega_n]^\omega$ in ZFC, and any $[\kappa]^\omega$ ($\kappa$ an infinite cardinal) assuming $V = L$.