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?

**Downloads:**

Slides from the workshop will be collected at the following page.

0 likes