  Given sets $X,Y$ and a regular cardinal $\mu$, let $\Phi(X,Y,\mu)$ be the
statement that for any function $f : X \times Y \to \mu$, there are functions
$g_1 : X \to \mu$ and $g_2 : Y \to \mu$ such that for all $(x,y) \in X \times
Y$, $$f(x,y) \le \max \{ g_1(x), g_2(y) \}.$$ In ZFC, the statement
$\Phi(\omega_1, \omega_1, \omega)$ is false. However, we show the theory ZF +
"the club filter on $\omega_1$ is normal" + $\Phi(\omega_1, \omega_1, \omega)$
(which is implied by ZF + DC + "$V = L(\mathbb{R})$" + "$\omega_1$ is
measurable") implies that for every $\alpha < \omega_1$ there is a $\kappa \in
(\alpha,\omega_1)$ such that in some inner model, $\kappa$ is measurable with
Mitchell order $\ge \alpha$.
