Some classes of finite homomorphism-homogeneous point-line geometries

Abstract

In 2006, P. J. Cameron and J. Nešetřril introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we classify finite homomorphism-homogeneous point-line geometries up to a certain point. We classify all disconnected point-line geometries, and all connected point-line geometries that contain a pair of intersecting proper lines (we say that a line is proper if it contains at least three points). In a way, this is the best one can hope for, since a recent result by Rusinov and Schweitzer implies that there is no polynomially computable characterization of finite connected homomorphism-homogeneous point-line geometries that do not contain a pair of intersecting proper lines (unless P=coNP).