Bounding 2d functions by products of 1d functions

Abstract

Given sets $X,Y$ and a regular cardinal μ, 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 $(x,y)\in X\times Y$, $f(x,y)\le \max \{g_1\left(x\right),g_2\left(y\right)\}$. In $\mathrm{ZFC}$, the statement $\Phi ({\omega }_1,{\omega }_1,\omega )$ is false. However, we show the theory $\mathrm{ZF}+\text{“the}\text{club}\text{filter}\text{on}{\omega }_1\text{is}\text{normal”}+\Phi ({\omega }_1,{\omega }_1,\omega )$ (which is implied by $\mathrm{ZF}+\mathrm{DC}$+ “$V=L\left(R\right)$” + “ω₁ is measurable”) implies that for every there is a $\kappa \in (\alpha ,{\omega }_1)$ such that in some inner model, κ is measurable with Mitchell order $\ge \alpha$.