Partial Linear Space

Definition

A partial linear space is a non-empty set of points, provided with a collection of subsets called lines such that any pair of points is contained in at most one line and every line contains at least two points.

A partial linear space which is not a graph or a linear space is called proper.

Definition

Suppose that $G$ is an automorphism group of a proper partial linear space. If $G$ is transitive on the ordered pairs of collinear points, as well as on the ordered pairs of non-collinear points, $G$ is called 2-ultrahomogeneity.

Lemma

If $G$ is an automorphism group of a proper partial linear space $S=(\mathcal{P},\mathcal{L})$ acting $2$-ultrahomogeneously on it, then $G$ is a rank $3$ permutation group on $\mathcal{P}$.

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Some relevant literatures:

• Partial linear spaces with a rank 3 affine primitive group of automorphisms
• A classification of finite partial linear spaces with a primitive rank 3 automorphism group of almost simple type
• Transitive decompositions of graph products - rank 3 grid type