Quasi-symmetric designs with block intersection numbers x and y are investigated. Let Γ be the usual block graph of such a design. Let be the number of triangles on any edge of , the complement of Γ. It is shown that for fixed values of x, y ⩾ 2 and there are only finitely many such designs. This extends earlier results about quasi-symmetric designs with special properties. We show connections with strongly resolvable designs and also with designs considered by Holliday. The symbolic calculations were carried out using MACSYMA