Flag-transitive $2$-$(v,k,\lambda )$ designs with $\lambda \ge (r,\lambda {)}^2$

Abstract

This paper is devoted to the study of $2$-designs with $\lambda \ge (r,\lambda {)}^2$ admitting a flag-transitive automorphism group $G$. The group $G$ has been shown to be point-primitive of either almost simple or affine type. In this paper, we classify the $2$-designs with $\lambda \ge (r,\lambda {)}^2>1$ admitting a flag-transitive almost simple automorphism group with socle ${\mathrm{PSL}}_n(q)$ or ${\mathrm{PSU}}_n(q)$ for $n\ge 3$.