The Galvin-Hajnal Theorem

Abstract

The so-called singular cardinal problem consists of the description of the possible size of the cardinal, \(X_\eta ^{cf\left( {{N_\eta }} \right)}\), that is ℶ(ℵ η ), the value of the gimel function at the argument ℵ η , for singular cardinals ℵ η . An estimate for this cardinal power is given by the Galvin-Hajnal theorem if ℵ η is an ℵ0-strong singular cardinal with uncountable cofinality. The centre of our investigations will be the Galvin-Hajnal formula, from which all other results on cardinals in this chapter will follow. For the first time it turns out that a profound cardinal property is a source of cardinal arithmetic.