The subject of Finite Geometries is a central part of discrete mathematics. Structures from Finite Geometries have led to solutions of open problems in extremal combinatorics, design theory, and coding theory and cryptography. However, there are no recent textbooks in finite geometry that have a broad scope. This book discusses non-Desarguesian planes and explains the recent proof techniques using polynomials in case of Desarguesian planes. Some applications in extremal combinatorics are also mentioned.
Introduction. Collineations. Coordinatization. Higher dimensional spaces. Arcs, ovals, and hyperovals. Blocking sets. (k,n)-arcs and multiple blocking sets. Complete arcs. Polarities in PG(n,q). Quadratic surfaces, Hermitian varieties. Arcs and caps in higher dimensional spaces. Higher dimensional representations. Generalized quadrangles, Möbius planes. Applications to graph theory and extremal combinatorics. Applications to coding theory and cryptography.