On $\left(1,{\omega }_1\right)$\emph{-}weakly universal functions

Abstract

A function $U:{\left[{\omega }_1\right]}^2⟶\omega$ is called $\left(1,{\omega }_1\right)$\emph{-weakly universal }if for every function $F:{\left[{\omega }_1\right]}^2⟶\omega$ there is an injective function $h:{\omega }_1⟶{\omega }_1$ and a function $e:\omega ⟶\omega$ such that $F\left(\alpha ,\beta \right)=e\left(U\left(h\left(\alpha \right),h\left(\beta \right)\right)\right)$ for every $\alpha ,\beta \in {\omega }_1$. We will prove that it is consistent that there are no $\left(1,{\omega }_1\right)$\emph{-}weakly universal functions, this answers a question of Shelah and Steprāns. In fact, we will prove that there are no $\left(1,{\omega }_1\right)$\emph{-}weakly universal functions in the Cohen model and after adding ${\omega }_2$ Sacks reals side-by-side. However, we show that there are $\left(1,{\omega }_1\right)$\emph{-}weakly universal functions in the Sacks model. In particular, the existence of such graphs is consistent with $♣$ and the negation of the Continuum Hypothesis.