# Sh:1088

• Shelah, S., & Steprāns, J. (2021). Universal graphs and functions on \omega_1. Ann. Pure Appl. Logic, 172(8), Paper No. 102986, 43.
• Abstract:
It is shown to be consistent with various values of \mathfrak{b} and \mathfrak{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.
• Version 2021-04-28 (37p) published version (43p)
Bib entry
@article{Sh:1088,
author = {Shelah, Saharon and Stepr{\={a}}ns, Juris},
title = {{Universal graphs and functions on {$\omega_1$}}},
journal = {Ann. Pure Appl. Logic},
fjournal = {Annals of Pure and Applied Logic},
volume = {172},
number = {8},
year = {2021},
pages = {Paper No. 102986, 43},
issn = {0168-0072},
mrnumber = {4266242},
mrclass = {03E17 (03C30 03C50 03E35 03E65)},
doi = {10.1016/j.apal.2021.102986}
}