Projective geometries in exponentially dense matroids. II

Projective geometries in exponentially dense matroids. II

We show for each positive integer a that, if M is a minor-closed class of matroids not containing all rank-(a+ 1) uniform matroids, then there exists an integer c such that either every rank-r matroid in M can be covered by at most r rank-a sets, or M contains the GF(q)-representable matroids for some prime power q and every rank-r matroid inM can be covered by at most cq rank-a sets. In the la...

Projective geometries in dense matroids

We prove that, given integers l, q ≥ 2 and n there exists an integer α such that, if M is a simple matroid with no l + 2point line minor and at least αq elements, then M contains a PG(n− 1, q′)-minor, for some prime-power q′ > q.

Exponentially Dense Matroids

This thesis deals with questions relating to the maximum density of rank-n matroids in a minor-closed class. Consider a minor-closed class M of matroids that does not contain a given rank-2 uniform matroid. The growth rate function is defined by hM(n) = max (|M | : M ∈M simple, r(M) ≤ n) . The Growth Rate Theorem, due to Geelen, Kabell, Kung, and Whittle, shows that the growth rate function is ...

Projective Representations II. Generalized chain geometries

In this paper, projective representations of generalized chain geometries are investigated, using the concepts and results of [5]. In particular, we study under which conditions such a projective representation maps the chains of a generalized chain geometry Σ(F,R) to reguli; this mainly depends on how the field F is embedded in the ring R. Moreover, we determine all bijective morphisms of a ce...

Projective Representations II. Generalized chain geometries