Cherlin's conjecture on finite primitive binary permutation groups

Let $G$ be a permutation group on a set $\Omega$. For a positive integer $n$ and tuples $I=(\omega_1,\ldots,\omega_n)$ and $J$ in $\Omega^n$, write $I^g=(\omega_1^g,\ldots,\omega_n^g)$ and let $I_{ij}=(\omega_i,\omega_j)$ denote the corresponding 2-subtuple. The group $G$ is binary if, for every positive integer $n$ and all $I,J\in\Omega^n$, there exists $g\in G$ with $I^g=J$ if and only if, for every pair $i<j$, there exists $g_{ij}\in G$ with $I_{ij}^{g_{ij}}=J_{ij}$. A group is primitive if it acts transitively and preserves no nontrivial partition of $\Omega$. An affine orthogonal group is a group $V\rtimes \mathrm{O}(V)$ in which $V$ is a vector space over a finite field equipped with a non-degenerate anisotropic quadratic form, acting on itself by translations with complement the full orthogonal group $\mathrm{O}(V)$. Cherlin's conjecture. A finite primitive binary permutation group must be one of the following: a symmetric group $\operatorname{Sym}(n)$ acting naturally on $n$ elements; a cyclic group of prime order acting regularly on itself; or an affine orthogonal group $V\rtimes \mathrm{O}(V)$ of the specified type. The conjecture proposes a classification of finite primitive binary permutation groups. The terminology and the three families are discussed later in the paper; Cherlin proved the affine case, while the remaining cases were reduced to almost simple groups, so the general classification was still being completed in this work.