INFTY Final Conference, March 2014

I gave an invited talk at the INFTY Final Conference meeting, Bonn, March 4-7, 2014. [Curiosity: Georg Cantor was born March 3, 1845]

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?

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