Size functions of subgeometry-closed classes of representable combinatorial geometries
Let \(F_q\) be a finite field of characteristic \(q.\) A projective geometry \(PG(r, F_q)\) is the rank \(r\) matroid on the points of the \((r-1)\)-dimensional projective vector space over \(F_q.\) If we delete from \(PG(r, F_q)\) all the points of a hyperplane we obtain the affine geometry \(AG(r, F_q).\) Let \(ex_q(M; r)\) be the number of \(F_q\)-representable simple matroids of rank \(r\) that have no submatroid (restriction) isomorphic to a fixed simple matroid \(M.\) The purpose of this paper is to find \(ex_q(M; r)\) for several infinite families of \(M.\) In particular it is proved that \[ \lim_{r\to \infty} \frac {ex_2(AG(r, F_2); r)} {2^r-1}=0. \]
scientific article; zbMATH DE number 1533338