Title: Same Graph, Different Universe.
Abstract: In a paper from 1998, answering a question of Hajnal, Soukup proved that ZFC+GCH is consistent with the existence of two graphs G,H of size and chromatic number $\kappa=\omega_2$, whose product GxH is countably chromatic.
The consistency of the statement for cardinals $\kappa>\omega_2$ remained open up until recently, where we demonstrated that in Godel’s contsructible universe, this holds simultaneously for every successor cardinal $\kappa$.
The key idea is the construction of two graphs G and H of size $\kappa$, and two $(<\kappa)$-distributive notions of forcing $\mathbb P$ and $\mathbb Q$, such that:
(1) $L^\mathbb P\models chr(G)=\kappa, chr(H)=\omega$;
(2) $L^\mathbb Q\models chr(G)=\omega, chr(H)=\kappa$.
Motivated by the above, in this talk we shall address the following general question:
Given a fixed graph G in a fixed universe V, what are the possible values for chr(G) among all cofinality-preserving forcing extensions of V?