It is shown to be consistent with various values of $\mathfrak{b}$, $\mathfrak{d}$ and $2^{{\mathrm{\aleph }}_1}$ that there is a universal graph on ${\omega }_1$. Moreover, it is also shown that it is consistent that there is a universal graph on ${\omega }_1$ — in other words, a universal symmetric function from ${\omega }_1^2$ to 2 — but no such function from ${\omega }_1^2$ to ω. The method used relies on iterating well known reals, such as Miller and Laver reals, and alternating this with the PID forcing which adds no new reals. The last sections examine the question of set valued universal functions.

中文翻译：

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对通用图形和功能ω ₁

它被证明与的各种值一致 $\mathfrak{b}$， $\mathfrak{d}$ 和 ${\mathrm{2个}}^{{\mathrm{\aleph }}_{\mathrm{1个}}}$ 上面有一个通用图 ${\omega }_{\mathrm{1个}}$。此外，还表明在上有一个通用图是一致的${\omega }_{\mathrm{1个}}$ —换句话说， ${\omega }_{\mathrm{1个}}^{\mathrm{2个}}$ 到2-但没有这样的功能 ${\omega }_{\mathrm{1个}}^{\mathrm{2个}}$到ω。所使用的方法依赖于迭代众所周知的实数（例如Miller和Laver实数），并将其与PID强制交替使用，而PID强制不添加任何新的实数。最后几节讨论了集值通用函数的问题。