  We study the global existence of the parabolic-parabolic Keller-Segel system
in $\R^d , d \ge 2$. We prove that initial data of arbitrary size give rise to
global solutions provided the diffusion parameter $\tau$ is large enough in the
equation for the chemoattractant. This fact was observed before in the
two-dimensional case by Biler, Guerra \& Karch (2015) and Corrias, Escobedo \&
Matos (2014). Our analysis improves earlier results and extends them to any
dimension $d \ge 3$. Our size conditions on the initial data for the global
existence of solutions seem to be optimal, up to a logarithmic factor in
$\tau$, when $\tau>>1$: we illustrate this fact by introducing two toy models,
both consisting of systems of two parabolic equations, obtained after a slight
modification of the nonlinearity of the usual Keller-Segel system. For these
toy models, we establish in a companion paper [4] finite time blowup for a
class of large solutions.
