On the connectivity of infinite graphs

Abstract

Let ${\mu \geq \omega}$ be regular, assume the Generalized Continuum Hypothesis and the principle ${\square_\lambda}$ holds for every singular ${\lambda}$ with ${{\rm cf}(\lambda) \leq \mu}$. Let X be a graph with chromatic number greater than ${\mu^+}$. Then X contains a ${\mu}$-connected subgraph Y of X whose chromatic number is greater than ${\mu^+}$.