Partial geometries and their extensions

A. A. Makhnev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 54 (1999) no. 5, pp. 895-945

Résumé

This survey consists of three parts: partial geometries, diagram geometries, and extensions of partial geometries. In the first part we present the main definitions, all known examples of partial geometries, some characterization theorems, and also certain results on semipartial geometries. In the second part we give definitions related to geometries belonging to diagrams, a series of general results concerning these geometries, examples of rank 2 geometries that play the role of building blocks for geometries of higher rank, and also theorems on the structure of some flag-transitive rank 2 geometries. In the third part we give a survey of results characterizing extensions of partial geometries, namely, 2-designs and dual 2-designs, nets and dual nets, and also generalized quadrangles. Moreover, in the concluding subsection of this part we present some generalizations of the notion of extension and the corresponding characterization results.

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