**August 21**

**Dan Hathaway**,
University of Vermont

**A relative of ${\rm ZF} + {\rm DC} + `\omega_1\text{ is measurable'}$**

Let $\Phi$ be the statement that for any function $f: \omega_1 \times \omega_1 \to \omega$, there are functions $g_1, g_2 : \omega_1 \to \omega$ such that for all $(x,y) \in \omega_1 \times \omega_1$, we have $f(x,y) \le \text{max }\{g_1(x), g_2(y)\}$. We will show that $\Phi$ follows from ${\rm ZF} + {\rm DC} + `\omega_1\text{ is measurable'}$. On the other hand using core models, we will show that $\Phi + `\text{the club filter on }\omega_1\text{ is normal'}$ implies there are inner models with many measurable cardinals. We conjecture that $\Phi$ and ${\rm ZF} + {\rm DC} + `\omega_1\text{ is measurable'}$ have the same consistency strength. The research is joint with Francois Dorais at the University of Vermont.