Constructing flag-transitive, point-primitive 2-designs from complete graphs

In this paper, we study 2-designs D=(P,BSn), where P can be viewed as the edge set of the complete graph Kn, and B can be identified as the edge set of a subgraph of Kn. We give a necessary condition for Sn to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group Sn. As an application, all pairs (D,G) are determined, where D is a 2-(v,k,λ) design with gcd⁡(v−1,k−1)=3 or 4, and G is flag-transitive with Soc(G)=An for n≥5. Furthermore, we show that there are infinite flag-transitive, point-primitive 2-(v,k,λ) designs with gcd⁡(v−1,k−1)≤(v−1)1/2 and alternating socle An with v=(n2).