### 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. 171102 He promise that after his visit, the new theorem proved will be added ; as had been afer his visit in Israel

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