  We present a family of Bell inequalities for three parties and arbitrarily
many outcomes, which can be seen as a natural generalization of the Mermin Bell
inequality. For a small number of outcomes, we verify that our inequalities
define facets of the polytope of local correlations. We investigate the quantum
violations of these inequalities, in particular with respect to the Hilbert
space dimension. We provide strong evidence that the maximal quantum violation
can only be reached using systems with local Hilbert space dimension exceeding
the number of measurement outcomes. This suggests that our inequalities can be
used as multipartite dimension witnesses.
