The Siemons--Wagner property and Bradley's conjecture

A transitive permutation group $G\le \text{Sym}(\text{Pcal})$ is said to have the Siemons–Wagner property ${\star }_3$ if some three-point subset has a $G$-orbit longer than the orbit of every four-point subset containing it. In this paper we develop a structural theory of three-point orbits of finite $2$-transitive groups (orbit-length hierarchies, pointwise and setwise stabilizers, and their geometric realisations, including the unital and the ovoid), and we use it to completely settle Bradley’s conjecture: a finite primitive group of degree at least $8$ satisfying ${\star }_3$ is isomorphic to $\text{PSL}(2,7)$ or $\text{PGL}(2,7)$.