  Two overlapping bipartite binary input Bell inequalities cannot be
simultaneously violated as this would contradict the usual no-signalling
principle. This property is known as monogamy of Bell inequality violations and
generally Bell monogamy relations refer to trade-offs between simultaneous
violations of multiple inequalities. It turns out that multipartite Bell
inequalities admit weaker forms of monogamies that allow for violations of a
few inequalities at once. Here we systematically study monogamy relations
between correlation Bell inequalities both within quantum theory and under the
sole assumption of no signalling. We first investigate the trade-offs in Bell
violations arising from the uncertainty relation for complementary binary
observables, and exhibit several network configurations in which a tight
trade-off arises in this fashion. We then derive a tight trade-off relation
which cannot be obtained from the uncertainty relation showing that it does not
capture monogamy entirely. The results are extended to Bell inequalities
involving different number of parties and find applications in
device-independent secret sharing and device-independent randomness extraction.
Although two multipartite Bell inequalities may be violated simultaneously, we
show that genuine multi-party non-locality, as evidenced by a generalised
Svetlichny inequality, does exhibit monogamy property. Finally, using the
relations derived we reveal the existence of flat regions in the set of quantum
correlations.
