The long-line graph of a combinatorial geometry. II: Geometries representable over two fields of different characteristics
Let q be a power of a prime and let s be zero or a prime not dividing q. Then the number of points in a combinatorial geometry (or simple matroid of rank n which is representable over GF(q) and a field of characteristic s is at most \((q^{\nu}-q^{\nu -1})\left( \begin{matrix} n+1\\ 2\end{matrix} \right)-n,\) where \(\nu =2^{q-1}-1\). The proof uses the notion of the long-line graph introduced in an earlier paper [Q. J. Math. Oxf., II. Ser. 39, 223-234 (1988)].
scientific article; zbMATH DE number 4051648