  We investigate the smallest set of requirements for inducing non-Markovian
dynamics in a collisional model of open quantum systems. This is done by
introducing correlations in the state of the environment and analyzing the
divisibility of the quantum maps from consecutive time steps. Our model and
results serve as a platform for the microscopic study of non-Markovian behavior
as well as an example of a simple scenario of non-Markovianity with purely
contractive maps, i.e. with no back-flow of information between system and
environment.
