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

In this paper, we study 2-designs $D=(P,B^{S_n})$, where $P$ can be viewed as the edge set of the complete graph $K_n$, and B can be identified as the edge set of a subgraph of $K_n$. We give a necessary condition for $S_n$ to be flag-transitive, and then present three ways to construct such 2-designs admitting a flag-transitive, point-primitive automorphism group $S_n$. As an application, all pairs $(D,G)$ are determined, where $D$ is a 2-$(v,k,\lambda )$ design with $\gcd (v-1,k-1)=3$ or 4, and G is flag-transitive with $Soc\left(G\right)=A_n$ for $n\ge 5$. Furthermore, we show that there are infinite flag-transitive, point-primitive 2-$(v,k,\lambda )$ designs with $\gcd (v-1,k-1)\le {(v-1)}^{1/2}$ and alternating socle $A_n$ with $v=\left(\begin{array}{c}n\\ 2\end{array}\right)$.