Flag-transitive, point-imprimitive symmetric 2-(v, k, λ)designs with k >λ (λ−3)/2

Let D=(P,B) be a symmetric 2-(v,k,λ) design admitting a flag-transitive, point-imprimitive automorphism group G that leaves invariant a non-trivial partition Σ of P. Praeger and Zhou [42] have shown that, there is a constant k0 such that, for each B∈B and Δ∈Σ, the size of |B∩Δ| is either 0 or k0. In the present paper we show that, if k>λ(λ−3)/2 and k0⩾3, D is isomorphic to one of the known flag-transitive, point-imprimitive symmetric 2-designs with parameters (45,12,3) or (96,20,4).