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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