Hedetniemi's conjecture for uncountable graphs

Abstract

It is proved that in Gödel's constructible universe, for every infinite successor cardinal $\kappa$, there exist graphs $\mathcal{G}$ and $\mathcal{H}$ of size and chromatic number $\kappa$, for which the product graph $\mathcal{G}\times \mathcal{H}$ is countably chromatic.

In particular, this provides an affirmative answer to a question of Hajnal from 1985.