On $(1,\omega _{1})$-weakly universal functions
Volume 247 / 2019
Abstract
A function $U:[ \omega _{1} ] ^{2}\rightarrow \omega $ is called $( 1,\omega _{1} ) $-weakly universal if for every function $F:[ \omega _{1} ] ^{2}\rightarrow \omega $ there is an injective function $h:\omega _{1}\rightarrow \omega _{1}$ and a function $e:\omega \rightarrow \omega $ such that $F( \alpha ,\beta ) =e( U( h( \alpha ) ,h( \beta ) ) ) $ for all $\alpha ,\beta \in \omega _{1}$. We will prove that it is consistent that there are no $( 1,\omega _{1} ) $-weakly universal functions; this answers a question of Shelah and Steprāns. In fact, we will prove that there are no $( 1,\omega _{1} ) $-weakly universal functions in the Cohen model and after adding $\omega _{2}$ Sacks reals side-by-side. However, we show that there are $( 1,\omega _{1})$-weakly universal functions in the Sacks model. In particular, the existence of such graphs is consistent with $\clubsuit $ and the negation of the Continuum Hypothesis.
