ABSTRACT: In contrast to their classical counterparts, one-dimensional quantum spin systems are interesting, they have intriguing behaviour, and they are difficult to study. I will describe a family of systems with nearest-neighbour interactions and O(n) symmetry. Its ground state phase diagram is expected to have a rich structure. For large spins it is possible to prove the occurrence of dimerisation in an open domain. The proof involves a loop representation and the method of cluster expansions. Joint work with Jakob Bjornberg, Peter Muhlbacher, and Bruno Nachtergaele.