### Universal graphs and functions on $\omega_1$

by Shelah and Steprans. [ShSr:1088]

It is shown to be consistent with various values of mathfrak{b}
and {d} 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^2_1 to 2 -- but no such function from
omega^2_1 to omega . The method used relies on iterating well
know reals, such as Miller and Laver reals, and alternating this with
the PID forcing which adds no new reals.

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