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

TL;DR

This paper investigates the existence of certain weakly universal functions on uncountable sets, proving their non-existence in some models of set theory and their existence in others, thus highlighting their independence from standard axioms.

Contribution

It establishes the consistency of the non-existence of (1,ω₁)-weakly universal functions in specific models and demonstrates their existence in the Sacks model, answering a question by Shelah and Steprāns.

Findings

Non-existence in Cohen and Sacks models with side-by-side addition of ω₂ Sacks reals.

Existence in the Sacks model.

Consistency with old and negation of CH.

Abstract

A function $U:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ is called $\left( 1,\omega_{1}\right) $\emph{-weakly universal }if for every function $F:\left[ \omega_{1}\right] ^{2}\longrightarrow\omega$ there is an injective function $h:\omega_{1}\longrightarrow\omega_{1}$ and a function $e:\omega \longrightarrow\omega$ 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 Stepr\={a}ns. 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 $\clubsuit$ and the negation of the Continuum Hypothesis.